Differential equation

Question

Find all solutions of the equation

y' + 2y = b(x) , (−∞ < x < +∞)

where

1 − |x|, if |x| ≤ 1

b(x) = 0, if |x| > 1

Answer 1

One approach to this problem is the integrating factor method. If we define the integrating factor P(x) as the exponent of some antiderivative of the coefficient of the y(x) term in the ODE, we find in this problem

P(x) = e^2x

We can multiply both sides of the ODE by this factor and then recognize that we can rewrite the left hand side of the equation as

y'(x) + 2y(x) = d/dx [ P(x) * y(x) ]

Now the ODE looks like

d/dx [ P(x) * y(x) ] = P(x) * b(x)

If we take the indefinite integral of both sides, the fundamental theorem of calculus tells us that the left hand side of the ODE will become

d/dx [ P(x) * y(x) ] -> P(x) * y(x)

and the right hand side will be the indefinite integral of our integrating factor P(x) times the piecewise nonhomogeneous term b(x). This integral is a little bit involved since it requires some boundary matching, but it isn't too bad.

There are technically four intervals of interest in this indefinite integral with boundaries at x = -1, x = 0, and x = +1. The integration itself is pretty straightforward, although it requires the use of integration by parts. For the first interval, x < -1, we have that the integrand is identically zero, which indicates that the value of the integral is also identically zero here (plus a constant of integration!). This means that at the point x = -1, the function y(x) must be equal to zero (ignoring the overall constant of integration for now since it will be the same for all segments of the piecewise function).

The second interval is treated the same way. We must solve the indefinite integral

∫e^2x(1+x)dx = e^2x (1+2x) / 4 + C

Boundary matching tells us that when x = -1, the integral must equal zero, therefore the constant of integration for this segment must be

C = e^-2 / 4

The third interval produces

∫e^2x(1-x)dx = -1/2 -e^2x / 4 (2x - 3) + C

Again, the constant of integration is compared to the value of the previous segment evaluated at x = 0 to find

C = e^-2 / 4

coincidentally the same value as the previous segment. The final segment is found by evaluating the previous segment at x = +1, and since we know the derivative is zero here we know that its value is not changing in this final interval x > +1. Integrating and evaluating gives

-1/2 + e^-2 / 4 - e² / 4 (2 - 3) = [cosh(2) - 1] / 2

using the identity (e^x + e^-x) / 2 = cosh(x).

Recall that although we have fixed each segment relative to every other segment, we have not yet specified the overall constant of integration for the piecewise function. If we gather up each of the definite segments into a new piecewise function B(x), the ODE at this point looks like

y(x) * P(x) = B(x) + C

Therefore the family of solutions for this differential equation is found to be

y(x) = [ B(x) + C ] / P(x), or

y(x) = C e^-2x + { 0 , x < -1

{ (1+2x) / 4 + 1 / [ 4 e^2x+2 ] , -1 < x < 0

{ (3-2x) / 4 + ( 1 / [ 4 e² ] - 1/2 ) e^-2x , 0 < x < +1

{ [ cosh(2) - 1 ] e^-2x / 2 , x > +1

Note that without specifying an initial condition, this problem cannot be solved using the method of Laplace transforms.

I hope this helps! Feel free to contact me with any questions.