- #1
Susanne217
- 317
- 0
Hi
I have differential equation
[tex]y'' + p(x)\cdot x' + q(x)y = 0[/tex] which have two solutions [tex]y_1(x)[/tex] and [tex]y_2(x)[/tex] where [tex]y_1(x) \neq 0[/tex]
show that [tex]y_2(t) = y_1(t)\int_{t_0}^{t} \frac{1}{y_1(s)^2} e^{-\int_{t_0}^s p(r) dr} ds[/tex] is also a solution.
I know that wronskian for this is defined
[tex]w(y_1,y_2)(t) = c \cdot e^{-\int p(t) dt}[/tex] according to Abels identity theorem.
My question do I use this property of theorem and properties of the Wronskian to derive c?
Homework Statement
I have differential equation
[tex]y'' + p(x)\cdot x' + q(x)y = 0[/tex] which have two solutions [tex]y_1(x)[/tex] and [tex]y_2(x)[/tex] where [tex]y_1(x) \neq 0[/tex]
show that [tex]y_2(t) = y_1(t)\int_{t_0}^{t} \frac{1}{y_1(s)^2} e^{-\int_{t_0}^s p(r) dr} ds[/tex] is also a solution.
Homework Equations
I know that wronskian for this is defined
[tex]w(y_1,y_2)(t) = c \cdot e^{-\int p(t) dt}[/tex] according to Abels identity theorem.
The Attempt at a Solution
My question do I use this property of theorem and properties of the Wronskian to derive c?
Last edited: