Civil Engineering (CE) Exam  >  Civil Engineering (CE) Notes  >  Engineering Mathematics  >  Solution of Differential Equation

Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) PDF Download

Initial and Boundary value problems


With initial value problems we had a differential equation and we specified the value of the solution and an appropriate number of derivatives at the same point (collectively called initial conditions). For instance for a second order differential equation the initial conditions are,
y(t0) = y
y'(t0) = y1
With boundary value problems we will have a differential equation and we will specify the function and/or derivatives at different points, which we’ll call boundary values. For second order differential equations, which will be looking at pretty much exclusively here, any of the following can, and will, be used for boundary conditions.
y(x0) = y0    y(x1)= y1
y’(x0) = y0    y’(x1) = y1
y’(x0) = y0    y(x1) = y1
y(x0) = y0    y’(x1) = y1
So, for the purposes of our discussion here we’ll be looking almost exclusively at differential equations in the form,
y'+ p (x) y'+ q(x)y = g(x)
In the earlier chapters, we said that a differential equation was homogeneous if
g(x) = 0
for all x.  Here we will say that a boundary value problem is homogeneous if in addition to g(x) = 0Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) we also have y0 = 0 
and y1 = 0 (regardless of the boundary conditions we use).  If any of these are not zero we will call the BVP non-homogeneous

It is important to now remember that when we say homogeneous (or nonhomogeneous) we are saying something not only about the differential equation itself but also about the boundary conditions as well.
When solving linear initial value problems a unique solution will be guaranteed under very mild conditions. With boundary value problems we will often have no solution or infinitely many solutions even for very nice differential equations that would yield a unique solution if we had initial conditions instead of boundary conditions.
So, with some of the basic stuff out of the way let’s find some solutions to a few boundary value problems. Note as well that there really isn’t anything new here yet.  We know how to solve the differential equation and we know how to find the constants by applying the conditions. The only difference is that here we’ll be applying boundary conditions instead of initial conditions.

Example 1: Solve the following BVP.
y' + 4y = 0     y(0) = -2    y(π / 4) = 1
Solution:
Okay, this is a simple differential equation so solve and so we’ll leave it to you to verify that the general solution to this is,
y(x) = c1cos(2x) + csin (2x)
Now all that we need to do is apply the boundary conditions.
-2 = y(0) =c1
1 = y (π / 4) =c2
The solution is then,
y(x) = -2cos(2x) +  sin(2x)
We mentioned above that some boundary value problems can have no solutions or infinite solutions we had better do a couple of examples of those as well here.  This next set of examples will also show just how small of a change to the BVP it takes to move into these other possibilities.

Example 2: Solve the following BVP.
y' + 4y = 0   y(0) = -2   y(2π) = -2
Solution:
We’re working with the same differential equation as the first example so we still have,
y(x) = c1cos(2x) + c2 sin(2x)
Upon applying the boundary conditions we get,
-2 = y(0) = c1
-2 = y(2π) = c1
So in this case, unlike previous example, both boundary conditions tell us that we have to have c1 = 2 and neither one of them tell us anything about
Remember however that all we’re asking for is a solution to the differential equation that satisfies the two given boundary conditions and the following function will do that,
y(x) = -2cos(2x) + c2sin(2x)
In other words, regardless of the value of c2 we get a solution and so, in this case we get infinitely many solutions to the boundary value problem.

Question for Solution of Differential Equation
Try yourself:Which of the following statements is true about boundary value problems (BVPs)?
View Solution

Laplace Transform

  1. Definition and Basic Properties
    Initial value problem algebra problem solution of the algebra problem
    solution of the initial value problem
    (i) Definition (Laplace Transform): The Laplace transform Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE), for all s such that this integral converges.
    Examples:
    Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE), Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).  Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (ii) Table of Laplace transform of functionsSolution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)(iii) Theorem (Linearity of the Laplace transform): Suppose Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) and Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) are defined for s > a, and and β are real numbers.  
    Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for s > a.
    (iv) Definition (Inverse Laplace transform): Given a function G, a function g such that Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) is called an inverse Laplace transform of G. In this event, we write Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (v) Theorem (Lerch): Let f and g be continuous on [0, ∞) and suppose that Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE). Then f = g.
    (vi) Theorem: If Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) and Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) and Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) and Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) are real numbers,
    then
    Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
  2. Solution of initial value problems using the laplace transform
    (i) Theorem (Laplace transform of a derivative): Let f be continuous on [0, ∞)  and suppose f' is piecewise continuous on [0, k] for every positive k.  Suppose also that Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) if s > 0.  Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (ii) Theorem (Laplace transform of a higher derivative): 
    Suppose f, f',...fn - 1 are continuous on [0, ∞)  and Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) is piecewise continuous on [0, k] for every positive k.  Suppose also that Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)and for j = 1, 2, ...., n - 1. Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    Examples: Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
  3. Convolution
    (i) Definition (Convolution): If f and g are defined on [0, ∞), then the convolution f * g of f with g is the function defined by Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for t ≥ 0.
    (ii) Theorem (Convolution theorem): If f * g is defined, then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (iii) Theorem: Let Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) and Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).  Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    Example: Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    Determine f such that Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (iv) Theorem: If Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) is defined, so is Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE), and Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    Example: Solve Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
  4. Unit impulses and the dirac’s delta function
    (i) Dirac’s delta function:Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) where;Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE); Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (ii) Theorem (Filtering property): Let a > 0 and let f be integrable on [0, ∞) and continuous at a.  Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
    Let theSolution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) definition of the Laplace transformation of the delta function.
    Example: Solve Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (iii) Laplace Transform Solution of Systems:
    Example: Solve the system
    Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
    (iv) Differential Equations with Polynomial Coefficients
    (a) Theorem: Let Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for s > b and suppose that F is differentiable.  Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for s > b.
    (b) Corollary: Let Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for s > b and let n be a positive integer.  Suppose F is n times differentiable.  Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for s > b.
    Example: Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    (c) Theorem: Let f be piecewise continuous on Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for every positive number k and suppose there are numbers M and b such that Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) for t ≥ 0. Let Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).  Then Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).
    Example: Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE).

The Wave equation

Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)   - one dimensional wave equation.
The variable t has the significance of time, the variable x is the spatial variable. Unknown function u(x,t) depends both of x and t. For example, in case of vibrating string the function u(x,t) means the string deviation from equilibrium in the point x at the moment t.
Consider the wave equation for the vibrating string in more details.
Applying the 2d Newton’s law to the portion of the string between points x and x → Δx we get the equation
Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) 
and dividing by the Δx and taking the limit Δx → o we obtain the wave equation
Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)     (1)
If both ends of the string are fixed then the boundary conditions are
u(0, t) = 0, u(1, t) = 0
and the initial conditions are In case of infinite string - ∞ < x < ∞, t > 0 the wave equation with initial conditions has solution (D’Alembert’s solution):
Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)

The Diffusion (or heat) Equation

The equation for this distribution is
Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
Now we consider the temperature distribution Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE), which depends on the point x of the rod and time t.
Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
In case of diffusion equation r(x)=0.
Initial temperature distribution u(x, 0) = f(x).
If the function f(x) is so-called delta function (at initial moment we have the heat source at one point with coordinate ξ) then the homogeneous (r(x) = 0) heat equation
Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
has the solution. Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
This is the so-called fundamental solution.

Example:  The Braselton (chemical reaction system with two components) on the unit interval.
Solution:
Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)
x ∈ [0,1]; D1, D2, B, C > 0
u(0, t) = u(1, t) = C, v(0, t) = v(1, t) = B / C

Question for Solution of Differential Equation
Try yourself:Which theorem is used in the solution of initial value problems using the Laplace transform for higher derivatives?
View Solution

The document Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
All you need of Civil Engineering (CE) at this link: Civil Engineering (CE)
65 videos|120 docs|94 tests

Up next

65 videos|120 docs|94 tests
Download as PDF

Up next

Explore Courses for Civil Engineering (CE) exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Download the FREE EduRev App
Track your progress, build streaks, highlight & save important lessons and more!
Related Searches

past year papers

,

Previous Year Questions with Solutions

,

Sample Paper

,

Free

,

Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)

,

shortcuts and tricks

,

Extra Questions

,

Important questions

,

Summary

,

MCQs

,

Semester Notes

,

Viva Questions

,

Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)

,

Solution of Differential Equation | Engineering Mathematics - Civil Engineering (CE)

,

study material

,

Exam

,

practice quizzes

,

mock tests for examination

,

pdf

,

ppt

,

video lectures

,

Objective type Questions

;