# Math Help - initial value problem O D E

1. ## initial value problem O D E

I have read the theory but I don't understand how to use is.
Don't know where to put the 2 before y'' when I'm solving and can't really solve it. If you have an hyperlink with more explanations and examples of solutions of ODE's I would also appreciate it.
Thanks!

$2y''+3y'+y=x^{2}+3\sin x , y(0)=12.5 , y'(0)=-6$

2. $
2y''+3y'+y=x^{2}+3\sin x , y(0)=12.5 , y'(0)=-6
$
Solve the homogeneous equation first. ie solve:

$2\lambda^2+3\lambda+1=0$

This can be done using the auxiliay equation method.

Next, you will need to choose a complimentary function. I would try something like $y=Ax^2+3sin(x)+3cos(x)$, but these questions have a habit of being a little tricky. You may need to modify this =S

Finally, you can just substitute the given values back into the equation. For $y'(0)=-6$ you will have to differentiate your general solution.
Normally,the two initial conditions give two simultaneous equations for A and B.

Two simultaneous equations and two unknowns ftw!

Anyway, post back how you get on.

3. Labda can be -1, but also -1/2
How can I use both? Or do I have to chose?

4. Originally Posted by Showcase_22
Solve the homogeneous equation first. ie solve:

$2\lambda^2+3\lambda+1=0$
Originally Posted by miepie
Labda can be -1, but also -1/2
How can I use both? Or do I have to chose?
The characteristic equation that Showcase posted comes from looking for a slution of the complementary equation

$2y'' + 3y' + y = 0$,

in the form

$y = e^{\lambda x}$.

As you mention, there are two solutions and thus, the solution of the complementary equation is

$y = c_1 e^{-\frac{1}{2}x} + c_2 e^{-x}$ (you want both).