Chapter 2.3, Problem 31E

Discontinuous Coefficients. As we will see in Chapter 3, occasions arise when the coefficients $P\left(x\right)$ in a linear equation fails to be continuous because of jump discontinuities. Fortunately, we may still obtain a “reasonable” solution. For Example, consider the initial value problem $\frac{dy}{dx}+P\left(x\right)y=x$, $y(0)=1$,

Where

$P\left(x\right)=\{1,0\le x\le 2,3,x>2$.

a. Find the general solution for $0\le x\le 2$.

b. Choose the constant in the solution of part (a) so that the initial condition is satisfied.

c. Find the general solution for $x>2$.

d. Now choose the constant in the general solution from part (c) so that the solution from part (b) and the solution from part (c) agree at $x=2$.By patching the two solutions together, we can obtain a continuous function that satisfies the differential equation except at $x=2$, where its derivative is undefined.

e. Sketch the graph of the solution from $x=0$ to $x=5$.