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 Page 1


 
 
PARTIAL DIFFERENTIAL EQUATIONS 
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either 
by  eliminating the  arbitrary  constants  or  by  eliminating  the  arbitrary  functions  from  the  given 
relation. 
The Physical system contains arbitrary constants or arbitrary functions or both. 
Equations which contain one or more partial derivatives are called Partial Differential Equations. 
Therefore, there must be atleast two independent variables and one dependent variable. 
Let us take  to be two independent variables and   to be dependent variable. 
Order: The Order of a partial differential equation is the order of the highest partial derivative in 
the equation. 
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE 
Notations 
  ,        ,      ,     ,      
Formation of Partial Differential Equation 
 Formation of PDE by elimination of Arbitrary Constants 
 Formation of PDE by elimination of Arbitrary Functions 
Solution of a Partial Differential Equation 
Let us consider a Partial Differential Equation of the form                  1 
If it is Linear in   and  , it is called a Linear Partial Differential Equation  
(i.e. Order and Degree of    and   is one) 
If it is Not Linear in   and  , it is called as nonlinear Partial Differential Equation 
(i.e. Order and Degree of    and   is other than one) 
Consider a relation of the type   
By  eliminating  the  arbitrary  constants   and   from  this  equation,  we  get , 
which is called a complete integral or complete solution of the PDE. 
A solution of  obtained by giving particular  values to  and   in the complete 
Integral is called a particular Integral. 
 
Page 2


 
 
PARTIAL DIFFERENTIAL EQUATIONS 
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either 
by  eliminating the  arbitrary  constants  or  by  eliminating  the  arbitrary  functions  from  the  given 
relation. 
The Physical system contains arbitrary constants or arbitrary functions or both. 
Equations which contain one or more partial derivatives are called Partial Differential Equations. 
Therefore, there must be atleast two independent variables and one dependent variable. 
Let us take  to be two independent variables and   to be dependent variable. 
Order: The Order of a partial differential equation is the order of the highest partial derivative in 
the equation. 
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE 
Notations 
  ,        ,      ,     ,      
Formation of Partial Differential Equation 
 Formation of PDE by elimination of Arbitrary Constants 
 Formation of PDE by elimination of Arbitrary Functions 
Solution of a Partial Differential Equation 
Let us consider a Partial Differential Equation of the form                  1 
If it is Linear in   and  , it is called a Linear Partial Differential Equation  
(i.e. Order and Degree of    and   is one) 
If it is Not Linear in   and  , it is called as nonlinear Partial Differential Equation 
(i.e. Order and Degree of    and   is other than one) 
Consider a relation of the type   
By  eliminating  the  arbitrary  constants   and   from  this  equation,  we  get , 
which is called a complete integral or complete solution of the PDE. 
A solution of  obtained by giving particular  values to  and   in the complete 
Integral is called a particular Integral. 
 
 
 
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER 
A  Differential  Equation  which  involves  partial  derivatives   and   only  and  no  higher  order 
derivatives  is  called  a  first  order  equation.  If     and   have  the  degree  one,  it  is  called  a  linear 
partial differential equation of first order; otherwise it is called a  non-linear partial equation of 
first order. 
Ex: 1)   is a linear Partial Differential Equation. 
      2)   is a non-linear Partial Differential Equation. 
LAGRANGE’S LINEAR EQUATION 
A  linear  Partial  Differential  Equation  of  order  one,  involving  a  dependent  variable   and  two 
independent  variables    and ,  and  is  of  the  form ,  where   are  functions  of 
 is called Lagrange’s Linear Equation. 
Solution of the Linear Equation 
Consider  
Now,  
Case  1:  If  it  is  possible  to  separate  variables  then,  consider  any  two  equations,  solve  them  by 
integrating. Let the solutions of these equations are  
 is the required solution of given equation. 
Case 2: If it is not possible to separate variables then 
  
To solve above type of problems we have following methods 
 Method  of  grouping:  In  some  problems,  it  is  possible  to  solve  any  two  of  the  equations 
    (or)         (or)    
In  such cases,  solve  the  differential equation,  get  the  solution  and  then  substitute  in the 
other differential equation 
 Method of Multiplier: consider  
In  this,  we  have  to  choose   so  that  denominator=0.  That  will  give  us  solution  by 
integrating . 
Page 3


 
 
PARTIAL DIFFERENTIAL EQUATIONS 
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either 
by  eliminating the  arbitrary  constants  or  by  eliminating  the  arbitrary  functions  from  the  given 
relation. 
The Physical system contains arbitrary constants or arbitrary functions or both. 
Equations which contain one or more partial derivatives are called Partial Differential Equations. 
Therefore, there must be atleast two independent variables and one dependent variable. 
Let us take  to be two independent variables and   to be dependent variable. 
Order: The Order of a partial differential equation is the order of the highest partial derivative in 
the equation. 
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE 
Notations 
  ,        ,      ,     ,      
Formation of Partial Differential Equation 
 Formation of PDE by elimination of Arbitrary Constants 
 Formation of PDE by elimination of Arbitrary Functions 
Solution of a Partial Differential Equation 
Let us consider a Partial Differential Equation of the form                  1 
If it is Linear in   and  , it is called a Linear Partial Differential Equation  
(i.e. Order and Degree of    and   is one) 
If it is Not Linear in   and  , it is called as nonlinear Partial Differential Equation 
(i.e. Order and Degree of    and   is other than one) 
Consider a relation of the type   
By  eliminating  the  arbitrary  constants   and   from  this  equation,  we  get , 
which is called a complete integral or complete solution of the PDE. 
A solution of  obtained by giving particular  values to  and   in the complete 
Integral is called a particular Integral. 
 
 
 
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER 
A  Differential  Equation  which  involves  partial  derivatives   and   only  and  no  higher  order 
derivatives  is  called  a  first  order  equation.  If     and   have  the  degree  one,  it  is  called  a  linear 
partial differential equation of first order; otherwise it is called a  non-linear partial equation of 
first order. 
Ex: 1)   is a linear Partial Differential Equation. 
      2)   is a non-linear Partial Differential Equation. 
LAGRANGE’S LINEAR EQUATION 
A  linear  Partial  Differential  Equation  of  order  one,  involving  a  dependent  variable   and  two 
independent  variables    and ,  and  is  of  the  form ,  where   are  functions  of 
 is called Lagrange’s Linear Equation. 
Solution of the Linear Equation 
Consider  
Now,  
Case  1:  If  it  is  possible  to  separate  variables  then,  consider  any  two  equations,  solve  them  by 
integrating. Let the solutions of these equations are  
 is the required solution of given equation. 
Case 2: If it is not possible to separate variables then 
  
To solve above type of problems we have following methods 
 Method  of  grouping:  In  some  problems,  it  is  possible  to  solve  any  two  of  the  equations 
    (or)         (or)    
In  such cases,  solve  the  differential equation,  get  the  solution  and  then  substitute  in the 
other differential equation 
 Method of Multiplier: consider  
In  this,  we  have  to  choose   so  that  denominator=0.  That  will  give  us  solution  by 
integrating . 
 
 
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER 
 A partial differential equation which involves first order partial derivatives   and   with degree 
higher than one and the products of   and   is called a non-linear partial differential equation. 
There are six types of non-linear partial differential equations of first order as given below. 
 Type I:  
 Type II:  
 Type III:          (variable separable method) 
 Type IV: Clairaut’s Form 
 Equation reducible to standard forms  and   and  
 and  
 CHARPIT’S METHOD 
Let us see in detail about these types. 
Type I:  
Equations of the type  i.e. equations containing   and   only 
Let the required solution be  
 and  
Substituting these values in , we get  
From this, we can obtain   in terms of    (or)   in terms of   
Let , then the required solution is  
Note: Since, the given equation contains two first order partial derivatives , the final 
solution should contain only two constants. 
Type II:  
Let us consider the Equations of the type                 1 
Let   is a function of    and  
i.e.  and  
Now,   .1  
            . a a  
1  is the 1
st
 order differential equation in terms of dependent variable   
and independent variable  . 
Solve this differential equation and finally substitute  gives the required solution. 
Page 4


 
 
PARTIAL DIFFERENTIAL EQUATIONS 
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either 
by  eliminating the  arbitrary  constants  or  by  eliminating  the  arbitrary  functions  from  the  given 
relation. 
The Physical system contains arbitrary constants or arbitrary functions or both. 
Equations which contain one or more partial derivatives are called Partial Differential Equations. 
Therefore, there must be atleast two independent variables and one dependent variable. 
Let us take  to be two independent variables and   to be dependent variable. 
Order: The Order of a partial differential equation is the order of the highest partial derivative in 
the equation. 
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE 
Notations 
  ,        ,      ,     ,      
Formation of Partial Differential Equation 
 Formation of PDE by elimination of Arbitrary Constants 
 Formation of PDE by elimination of Arbitrary Functions 
Solution of a Partial Differential Equation 
Let us consider a Partial Differential Equation of the form                  1 
If it is Linear in   and  , it is called a Linear Partial Differential Equation  
(i.e. Order and Degree of    and   is one) 
If it is Not Linear in   and  , it is called as nonlinear Partial Differential Equation 
(i.e. Order and Degree of    and   is other than one) 
Consider a relation of the type   
By  eliminating  the  arbitrary  constants   and   from  this  equation,  we  get , 
which is called a complete integral or complete solution of the PDE. 
A solution of  obtained by giving particular  values to  and   in the complete 
Integral is called a particular Integral. 
 
 
 
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER 
A  Differential  Equation  which  involves  partial  derivatives   and   only  and  no  higher  order 
derivatives  is  called  a  first  order  equation.  If     and   have  the  degree  one,  it  is  called  a  linear 
partial differential equation of first order; otherwise it is called a  non-linear partial equation of 
first order. 
Ex: 1)   is a linear Partial Differential Equation. 
      2)   is a non-linear Partial Differential Equation. 
LAGRANGE’S LINEAR EQUATION 
A  linear  Partial  Differential  Equation  of  order  one,  involving  a  dependent  variable   and  two 
independent  variables    and ,  and  is  of  the  form ,  where   are  functions  of 
 is called Lagrange’s Linear Equation. 
Solution of the Linear Equation 
Consider  
Now,  
Case  1:  If  it  is  possible  to  separate  variables  then,  consider  any  two  equations,  solve  them  by 
integrating. Let the solutions of these equations are  
 is the required solution of given equation. 
Case 2: If it is not possible to separate variables then 
  
To solve above type of problems we have following methods 
 Method  of  grouping:  In  some  problems,  it  is  possible  to  solve  any  two  of  the  equations 
    (or)         (or)    
In  such cases,  solve  the  differential equation,  get  the  solution  and  then  substitute  in the 
other differential equation 
 Method of Multiplier: consider  
In  this,  we  have  to  choose   so  that  denominator=0.  That  will  give  us  solution  by 
integrating . 
 
 
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER 
 A partial differential equation which involves first order partial derivatives   and   with degree 
higher than one and the products of   and   is called a non-linear partial differential equation. 
There are six types of non-linear partial differential equations of first order as given below. 
 Type I:  
 Type II:  
 Type III:          (variable separable method) 
 Type IV: Clairaut’s Form 
 Equation reducible to standard forms  and   and  
 and  
 CHARPIT’S METHOD 
Let us see in detail about these types. 
Type I:  
Equations of the type  i.e. equations containing   and   only 
Let the required solution be  
 and  
Substituting these values in , we get  
From this, we can obtain   in terms of    (or)   in terms of   
Let , then the required solution is  
Note: Since, the given equation contains two first order partial derivatives , the final 
solution should contain only two constants. 
Type II:  
Let us consider the Equations of the type                 1 
Let   is a function of    and  
i.e.  and  
Now,   .1  
            . a a  
1  is the 1
st
 order differential equation in terms of dependent variable   
and independent variable  . 
Solve this differential equation and finally substitute  gives the required solution. 
 
 
Type III:          (variable separable method) 
Let us consider the differential equation is of the form          
Let  (say) 
Now             (I.e. writing   in terms of  ) 
              (I.e. writing   in terms of  ) 
Now,  
                  
   
By Integrating this, we get the required solution. 
Note: This method is used only when it is possible to separate variables.  
i.e.   on one side and  on other side. 
 
Type IV: Clairaut’s Form 
Equations of the form  
Let the required solution be , then 
  and       
   Required solution is  
i.e. Directly substitute   in place of   and   in place of   in the given equation. 
 
Equations Reducible to Standard Forms 
Equations of the type , where  and   are constants. 
Now, let us transform above equation to the form  (Type-I) 
Case-I: If  and  
Put  and  , then 
      ( and  ) 
        ( and  ) 
Substituting these values in the given equation, we get 
 
                                   
which is in the form of  (Type-I) 
Solve this, get the result which will be in terms of  and   and the substitute  and 
, which is the required solution. 
Page 5


 
 
PARTIAL DIFFERENTIAL EQUATIONS 
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either 
by  eliminating the  arbitrary  constants  or  by  eliminating  the  arbitrary  functions  from  the  given 
relation. 
The Physical system contains arbitrary constants or arbitrary functions or both. 
Equations which contain one or more partial derivatives are called Partial Differential Equations. 
Therefore, there must be atleast two independent variables and one dependent variable. 
Let us take  to be two independent variables and   to be dependent variable. 
Order: The Order of a partial differential equation is the order of the highest partial derivative in 
the equation. 
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE 
Notations 
  ,        ,      ,     ,      
Formation of Partial Differential Equation 
 Formation of PDE by elimination of Arbitrary Constants 
 Formation of PDE by elimination of Arbitrary Functions 
Solution of a Partial Differential Equation 
Let us consider a Partial Differential Equation of the form                  1 
If it is Linear in   and  , it is called a Linear Partial Differential Equation  
(i.e. Order and Degree of    and   is one) 
If it is Not Linear in   and  , it is called as nonlinear Partial Differential Equation 
(i.e. Order and Degree of    and   is other than one) 
Consider a relation of the type   
By  eliminating  the  arbitrary  constants   and   from  this  equation,  we  get , 
which is called a complete integral or complete solution of the PDE. 
A solution of  obtained by giving particular  values to  and   in the complete 
Integral is called a particular Integral. 
 
 
 
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER 
A  Differential  Equation  which  involves  partial  derivatives   and   only  and  no  higher  order 
derivatives  is  called  a  first  order  equation.  If     and   have  the  degree  one,  it  is  called  a  linear 
partial differential equation of first order; otherwise it is called a  non-linear partial equation of 
first order. 
Ex: 1)   is a linear Partial Differential Equation. 
      2)   is a non-linear Partial Differential Equation. 
LAGRANGE’S LINEAR EQUATION 
A  linear  Partial  Differential  Equation  of  order  one,  involving  a  dependent  variable   and  two 
independent  variables    and ,  and  is  of  the  form ,  where   are  functions  of 
 is called Lagrange’s Linear Equation. 
Solution of the Linear Equation 
Consider  
Now,  
Case  1:  If  it  is  possible  to  separate  variables  then,  consider  any  two  equations,  solve  them  by 
integrating. Let the solutions of these equations are  
 is the required solution of given equation. 
Case 2: If it is not possible to separate variables then 
  
To solve above type of problems we have following methods 
 Method  of  grouping:  In  some  problems,  it  is  possible  to  solve  any  two  of  the  equations 
    (or)         (or)    
In  such cases,  solve  the  differential equation,  get  the  solution  and  then  substitute  in the 
other differential equation 
 Method of Multiplier: consider  
In  this,  we  have  to  choose   so  that  denominator=0.  That  will  give  us  solution  by 
integrating . 
 
 
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER 
 A partial differential equation which involves first order partial derivatives   and   with degree 
higher than one and the products of   and   is called a non-linear partial differential equation. 
There are six types of non-linear partial differential equations of first order as given below. 
 Type I:  
 Type II:  
 Type III:          (variable separable method) 
 Type IV: Clairaut’s Form 
 Equation reducible to standard forms  and   and  
 and  
 CHARPIT’S METHOD 
Let us see in detail about these types. 
Type I:  
Equations of the type  i.e. equations containing   and   only 
Let the required solution be  
 and  
Substituting these values in , we get  
From this, we can obtain   in terms of    (or)   in terms of   
Let , then the required solution is  
Note: Since, the given equation contains two first order partial derivatives , the final 
solution should contain only two constants. 
Type II:  
Let us consider the Equations of the type                 1 
Let   is a function of    and  
i.e.  and  
Now,   .1  
            . a a  
1  is the 1
st
 order differential equation in terms of dependent variable   
and independent variable  . 
Solve this differential equation and finally substitute  gives the required solution. 
 
 
Type III:          (variable separable method) 
Let us consider the differential equation is of the form          
Let  (say) 
Now             (I.e. writing   in terms of  ) 
              (I.e. writing   in terms of  ) 
Now,  
                  
   
By Integrating this, we get the required solution. 
Note: This method is used only when it is possible to separate variables.  
i.e.   on one side and  on other side. 
 
Type IV: Clairaut’s Form 
Equations of the form  
Let the required solution be , then 
  and       
   Required solution is  
i.e. Directly substitute   in place of   and   in place of   in the given equation. 
 
Equations Reducible to Standard Forms 
Equations of the type , where  and   are constants. 
Now, let us transform above equation to the form  (Type-I) 
Case-I: If  and  
Put  and  , then 
      ( and  ) 
        ( and  ) 
Substituting these values in the given equation, we get 
 
                                   
which is in the form of  (Type-I) 
Solve this, get the result which will be in terms of  and   and the substitute  and 
, which is the required solution. 
 
 
Case-II: If  and  
Put  and  , then 
         ( and  ) 
        ( and  )  
Substituting these values in the given equation, we get 
 
                                   
                                    (Type-I) 
 Solve this, get the result which will be in terms of  and   and the substitute  and 
, which is the required solution. 
? Equations of the type , where  and   are constants 
This equation can be reduced in to   (Type-II) by taking above substitutions. 
Equations of the type , where    is a constant 
In order to convert into the form , we have to take the following substitutions 
Put  
? Equations of the type  where   is a constant. 
These type of equations can be reduced to the form   (Type-I)  (or)  
 by taking above substitutions given for the equation   
CHARPIT’S METHOD 
This  is  a  general  method  to  find  the  complete  integral  of  the  non-linear  PDE  of  the  form 
             
Now Auxillary Equations are given by 
 
Here we have to take the terms whose integrals are easily calculated, so that it may be easier to 
solve   and  . 
Finally substitute in the equation  
Integrate it, we get the required solution. 
Note that all the above (TYPES) problems can be solved in this method.
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FAQs on PPT: Partial Differential Equations - Engineering Mathematics - Civil Engineering (CE)

1. What are partial differential equations?
Ans. Partial differential equations (PDEs) are mathematical equations that involve multiple variables and their partial derivatives. They describe the relationships between multiple unknown functions and their partial derivatives with respect to the variables. PDEs are widely used in various fields of science and engineering to model and analyze physical phenomena.
2. How are partial differential equations different from ordinary differential equations?
Ans. The main difference between partial differential equations (PDEs) and ordinary differential equations (ODEs) is that PDEs involve multiple independent variables, while ODEs involve only one independent variable. PDEs describe functions that depend on multiple variables and their partial derivatives, whereas ODEs describe functions that depend only on one variable and its derivatives.
3. What are some common methods for solving partial differential equations?
Ans. There are several methods for solving partial differential equations (PDEs), depending on the type of equation and its properties. Some common methods include separation of variables, method of characteristics, Fourier series, Laplace transform, numerical methods (such as finite difference, finite element, and finite volume methods), and Green's functions. The choice of method depends on the specific problem at hand and the desired level of accuracy.
4. Can you provide an example of a practical application of partial differential equations?
Ans. Sure! One practical application of partial differential equations is in heat conduction analysis. The heat equation, which is a type of PDE, is used to model the distribution of heat in a solid material over time. By solving the heat equation, engineers can predict how heat will propagate through different materials, which is crucial for designing efficient heating or cooling systems, understanding thermal behavior in electronic devices, and optimizing energy consumption.
5. What is the importance of partial differential equations in the field of physics?
Ans. Partial differential equations (PDEs) play a fundamental role in the field of physics. They are used to describe and analyze physical phenomena, such as fluid flow, electromagnetic fields, quantum mechanics, and general relativity. PDEs provide mathematical models that help physicists understand and predict the behavior of complex physical systems. They are essential tools for formulating and solving problems in various branches of physics, including classical mechanics, fluid dynamics, electromagnetism, and quantum physics.
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Document Description: PPT: Partial Differential Equations for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation. The notes and questions for PPT: Partial Differential Equations have been prepared according to the Civil Engineering (CE) exam syllabus. Information about PPT: Partial Differential Equations covers topics like and PPT: Partial Differential Equations Example, for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for PPT: Partial Differential Equations.
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