Efficient Techniques for Solving Second Order Differential Equations

[nysnacc]

Homework Statement

Homework Equations

The Attempt at a Solution

Should I find the characteristic equation then find the solution y2? Or just integrate the expression of part (a)??

Part B, just find the derivatives and plug into the ode

 

[Ssnow]

Should I find the characteristic equation then find the solution y2? Or just integrate the expression of part (a)??

yes you can put the expression in the ODE and use the fact that $y_{1}$ is a particular solution ...

 

[nysnacc]

So, I calculate the y2, y2' and y2" in terms of y1, and plug into ODE (1st line)??

 

[Ssnow]

yes, as hint I show you how to derive $y_{2}(x)$ respect $x$. We observe that we have a product so:

$y_{2}'(x)=y_{1}'(x)\int \frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}dx + y_{1}(x)\frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}$

try for $y_{2}''$ ...

 

[nysnacc]

d/dx $y_{1}(x)\frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}$

= -1/y1^2 $e^{-\int p(x)dx}$ +1/y p(x) $e^{-\int p(x)dx}$

 

[nysnacc]

After hving the y, y', y", what should I do?

 

[Ssnow]

d/dx y1(x)e−∫p(x)dxy21(x)y_{1}(x)\frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}

= -1/y1^2 e^{-\int p(x)dx}}e^{-\int p(x)dx}} +1/y p(x) e^{-\int p(x)dx}}e^{-\int p(x)dx}}

wait this not $y_{2}''(x)$ this is only a part of the second derivative, you forgot to derive the term $y_{1}'(x)\int\frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}dx$ ...

 

[nysnacc]

Yes, you're right, but is this part correct tho?

 

[Ssnow]

yes but there is $-$ instead $+$ in the middle...

 

[nysnacc]

oh right, because of ∫ -p(x) :)

 

[nysnacc]

Then I plug into the ODE?? but I have 2 unknowns p(x) and q(x) ... what should I do?

 

[Ssnow]

If your calculation are correct you must have this:

$y_{1}''(x)\int \frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}dx +y_{1}'(x)\frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}+\frac{-p(x)y(x)e^{-\int p(x)dx}-y_{1}'(x)e^{-\int p(x)dx}}{y_{1}^{2}(x)}+$
$+p(x)y_{1}'\int \frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}dx+p(x)\frac{e^{-\int p(x)dx}}{y_{1}(x)}+q(x)y_{1}(x)\int \frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}dx=0$

 

[Ssnow]

that is $\left[y_{1}''(x)+p(x)y_{1}'(x)+q(x)y_{1}(x)\right]\int\frac{e^{-\int p(x)dx}}{y_{1}^{2}(x)}dx=0$, you can simplify and this prove the first part ...

 

[vela]

Then I plug into the ODE?? but I have 2 unknowns p(x) and q(x) ... what should I do?

You need to use the fact that $y_1$ is a solution of the ODE.

 

[nysnacc]

thank you