Solution Found!
By completing the following steps, prove that the
Chapter 4, Problem 32E(choose chapter or problem)
By completing the following steps, prove that the Wronskian of any two solutions \(y_{1}, y_{2}\) to the equation \(y^{\prime \prime}+p y^{\prime}+q y=0\) on \((a, b)\) is given by Abel's formula
\(W\left[y_{1}, y_{2}\right](t)=\operatorname{Cexp}\left\{-\int_{t_{0}}^{t} p(\tau) d \tau\right\}, t_{0}\) and \(t\) in \((a,b)\)
where the constant \(C\) depends on \(y_1\) and \(y_2\).
(a) Show that the Wronskian \(W\) satisfies the equation \(W^{\prime}+p W=0\)
(b) Solve the separable equation in part (a).
(c) How does Abel’s formula clarify the fact that the Wronskian is either identically zero or never zero on \((a,b)\)?
Questions & Answers
QUESTION:
By completing the following steps, prove that the Wronskian of any two solutions \(y_{1}, y_{2}\) to the equation \(y^{\prime \prime}+p y^{\prime}+q y=0\) on \((a, b)\) is given by Abel's formula
\(W\left[y_{1}, y_{2}\right](t)=\operatorname{Cexp}\left\{-\int_{t_{0}}^{t} p(\tau) d \tau\right\}, t_{0}\) and \(t\) in \((a,b)\)
where the constant \(C\) depends on \(y_1\) and \(y_2\).
(a) Show that the Wronskian \(W\) satisfies the equation \(W^{\prime}+p W=0\)
(b) Solve the separable equation in part (a).
(c) How does Abel’s formula clarify the fact that the Wronskian is either identically zero or never zero on \((a,b)\)?
ANSWER:Solution:
Step 1:
In this question, we have to show that the Wronskian of any two solutions y1, y2 to the equation
