Important Differential Equations Formulas for JEE and NEET

Differential Equations  Of  First  Order  And  First  Degree

Definitions :

1. An equation  that  involves  independent  and  dependent  variables  and  the  derivatives  of  the dependent variables  is  called  a  Differential Equations.

2. A differential  equation  is  said  to  be  ordinary ,  if  the  differential  coefficients  have  reference to  a single  independent  variable  only  and  it  is  said  to  be  PARTIAL   if  there  are  two  or more independent  variables .  We are concerned  with  ordinary  differential  equations  only. eg.

= 0  is a partial differential equation. 

3. Finding  the unknown function is called solving or integrating the differential  equation . The solution  of  the  differential  equation  is  also  called  its  Primitive, because  the  differential equation can  be  regarded  as  a  relation  derived  from  it. 

4. The  order  of  a differential equation  is  the  order  of  the  highest  differential  coefficient occuring  in  it. 

5. The  degree  of  a  differential  equation  which can be written as a polynomial in the derivatives  is  the degree of  the  derivative  of   the  highest  order  occuring  in  it , after  it  has  been  expressed  in a form free from radicals  &  fractions  so far  as  derivatives  are  concerned, thus  the  differential  equation :  

= 0  is  order  m  &  degree p. Note  that  in  the  differential equation    ey′′′ − xy′′ + y = 0    order  is  three  but  degree doesn't   apply. 

6. Formation of a differential equation : If  an  equation  in  independent  and  dependent  variables  having  some  arbitrary constant  is given ,  then a  differential  equation  is  obtained  as  follows: Differentiate  the  given  equation   w.r.t.   the  independent  variable  (say x)  as  many times as  the  number  of  arbitrary  constants  in  it . Eliminate  the  arbitrary  constants  .

The eliminant is the required  differential equation . Consider forming  a  differential equation  for  y² =  4a(x + b)  where  a  and  b  are  arbitary  constant .

Note : A differential equation represents a family of curves all satisfying some common properties. This can  be  considered  as  the  geometrical  interpretation  of  the differential equation. 

7. General And Particular Solutions : The solution of a differential equation which contains a number of independent arbitrary constants equal to the order of the differential equation is called the general solution (or complete integral or complete primitive) . A solution obtainable from the general solution by giving particular values to the constants is called a Particular solution.

Note that the general solution of a differential equation of the nth order contains ‘n’ & only ‘n’ independent arbitrary constants. The arbitrary constants in the solution of a differential equation are said to be independent, when it is impossible to deduce from the solution an equivalent relation containing fewer arbitrary constants. Thus the two arbitrary constants A, B in the equation  y = A ex + B are not independent since the equation can be written as  y = A e^B. ex = C ^ex. Similarly the solution y = A sin x + B cos (x + C) appears to contain three arbitrary constants, but they are really equivalent to two only. 

8. Elementary  Types  Of  First  Order  &  First  Degree  Differential  Equations.

TYPE−−−−1. Variables separable : If  the  differential  equation can be expressed as ; f (x)dx + g(y)dy = 0  then  this  is  said  to  be  variable − separable  type.

A  general  solution  of  this  is  given  by

where  c  is  the  arbitrary  constant  .  consider  the  example  (dy/dx) =  e^x−y + x². e^−y.

Note : Sometimes  transformation  to  the  polar  co−ordinates  facilitates  separation of variables.

In  this  connection  it  is  convenient  to  remember  the  following differentials. If  x = r cos θ ;  y = r sin θ  then, 

(i)  x dx + y dy = r dr 

(ii)  dx₂ + dy₂ = dr₂ + r2 dθ₂ 

(iii)  x dy − y dx = r₂ dθ If  x = r sec θ  &  y = r tan θ  then  x dx − y dy = r dr   and  x dy − y dx = r₂ sec θ dθ .

TYPE− 2 : 

To  solve  this ,  substitute  t = ax + by + c. Then  the  equation reduces  to  separable  type  in the  variable  t  and  x  which  can  be  solved. Consider  the

example

TYPE− 3.  Homogeneous equations

A differential  equation  of  the  form   where  f (x , y)  &  φ (x , y) are  homogeneous  functions of  x & y , and  of  the  same degree , is  called  Homogeneous .This  equation  may  also  be  reduced  to the  form

& is solved by putting  y = vx  so that the  dependent  variable  y  is  changed  to another variable  v, where v  is  some unknown function,  the  differential  equation  is  transformed to  an equation  with  variables  separable.  Consider

TYPE− 4.  Equations reducible to the homogeneous form:

If  where  a₁ b₂ − a₂ b₁ ≠  0,  

then  the  substitution  x = u + h,  y = v + k  transform  this  equation  to  a  homogeneous type in  the  new variables u  and  v  where  h  and  k  are  arbitrary  constants  to  be chosen so as to  make  the  given equation  homogeneous  which  can  be  solved  by  the method  as  given in  Type − 3. 

If (i) a₁ b₂ − a₂ b₁ = 0 ,  then  a  substitution  u = a₁ x + b₁ y  transforms  the  differential

equation  to an  equation with  variables  separable.   and 

(ii) b₁ + a₂ = 0 ,  then  a  simple  cross  multiplication  and  substituting  d (xy)  for  x dy + y dx  &  integrating term by term yields the result easily.

Consider

(iii) In an equation of the form :  yf (xy) dx + xg (xy)dy = 0  the variables can be separated by the substitution xy = v.

Important note :

(a) The  function  f (x , y)  is  said  to  be  a  homogeneous  function  of  degree  n  if  for  any real  number  t (≠ 0) ,  we  have  f (tx , ty) =  tn  f(x , y) .

For  e.g.  f(x , y)  =  ax^2/3 + hx^1/3 . y^1/3 + by^2/3 is a homogeneous function of degree 2/3

(b) A  differential equation  of  the  form  is  homogeneous  if  f(x , y) is a homogeneous function  of  degree zero   i.e.   f(tx , ty)  =  t° f(x , y) = f(x , y).  The function  f  does  not  depend on  x & y  separately  but  only  on  their  ratio

Linear diferential equations: A differential equation is said to be linear if the dependent variable  & its differential   coefficients occur in the first degree only and are not multiplied together The  nth  order  linear  differential  equation is  of  the  form  

coefficients  of  the  differential  equation. Note that a linear differential equation is always of  the first degree but every differental equation of  the   first degree need not be   linear. e.g. the differential equation  is not linear, though  its degree is 1.

TYPE −−−− 5.  Linear differential equations of first order :

The most general form of a linear differential equations of first order is  

 where P& Q  are  functions  of  x . To  solve  such  an  equation  multiply  both  sides  by

Note : (1) The factor  on multiplying by which the left hand side of the differential equation becomes  the differential coefficient  of  some  function  of  x  &  y ,  is called integrating factor of the differential equation popularly abbreviated as I. F.

(2) It is very important to remember that on multiplying by the integrating factor , the left  hand  side  becomes the  derivative  of  the  product  of  y  and  the  I. F. (3) Some times a  given differential  equation  becomes  linear  if  we  take  y  as  the independent variable and x  as  the  dependent  variable.   e.g.  the  equation  ; 

 which  is  a  linear differential

equation.

TYPE−−−−6.  Equations reducible to linear form  :

The  equation  yn   where  P & Q functions  of  x ,  is reducible to  the  linear form by dividing it by  yⁿ  &  then substituting  y⁻ⁿ⁺¹ = Z .  Its solution  can  be  obtained  as in Type−−−−5. Consider  the example  (x³ y² + xy) dx = dy.

The  equation yⁿ is  called  Bernouli’s equation.

9. Trajectories: Suppose we are given the family of plane curves. Φ (x, y, a) = 0 depending on a single parameter a. A curve making at each of its points a fixed angle α with the curve of the family passing through that point is called an isogonal trajectory of that family ; if in particular α = π/2, then it is called an orthogonal trajectory.

Orthogonal trajectories : We set up the differential equation of the given family of curves. Let it be of the form F (x, y, y') = 0 The differential equation of the orthogonal trajectories is of the form F   0 The general integral of this equation Φ1 (x, y, C) = 0 gives the family of orthogonal trajectories.

Note : Following  exact  differentials  must  be  remembered  :