Chapter 3.1, Problem 32P

Let $y_1\text{and}{y}_2$ be two solutions of $A\left(x\right)r”+B\left(x\right)y’+C\left(x\right)y=0$ on an open interval I where A, B, and C are continuous and A (x ) is never zero. (a) Let $W\left(y_1,y_2\right)$. Show that $A\left(x\right)\frac{dW}{dx}=\left(y_1\right)\left(Ay{"}_2\right)-\left(y_2\right)\left(A{y}_1"\right)$. Then substitute for $Ay{"}_2$ and $A{y}_1"$ from the original differential equation to show that $A\left(x\right)\frac{dW}{dx}=-B\left(x\right)W\left(x\right)$ .

(b) Solve this first-order equation to deduce Abel’s formula

$W\left(x\right)K\exp \left(-{\displaystyle \int \frac{B\left(x\right)}{A\left(x\right)}dx}\right),$ where K is a constant. (c) Why does Abel’s formula imply that the Wronskian $W\left(y_1,y_2\right)$ is either zero everywhere or nonzero everywhere (as stated in Theorem 3)?