Use the method of undetermined coefficients to solve the following differential equation: $\displaystyle y''+y'=4x$

$\displaystyle y(x)=$______$\displaystyle +C1+$______$\displaystyle C2$_____

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- Apr 15th 2008, 11:50 AMkillasnakeMethod of Undetermined Coefficients
Use the method of undetermined coefficients to solve the following differential equation: $\displaystyle y''+y'=4x$

$\displaystyle y(x)=$______$\displaystyle +C1+$______$\displaystyle C2$_____ - Apr 15th 2008, 02:48 PMKalter Tod
I'm a little rusty on Undetermined Coefficients, but you want to start by arbitrarily choosing a function in the same form as the inhomogeneous one you're giving.

For example, $\displaystyle y_{p}=ax+b$ should work for this situation.

Find the derivative, and second derivative so that

$\displaystyle y_{p}'=a$

and $\displaystyle y_{p}''=0$

Now that you have that, you can plug in these functions where they are needed in the original. So, you should get

$\displaystyle a+0=4x$

I'm sort of confused, and maybe you left out a y factor, but if this is right, then you get 0 for both coefficients.

So, assuming that you did write the original equation correctly, then the solution to the inhomogeneous part is as follows: $\displaystyle y_{p}(x)=0$

You can solve for $\displaystyle y_{h}(x)$ using your typical method of solving for $\displaystyle \lambda_{1}$ and $\displaystyle \lambda_{2}$

If you need help with the homogeneous solution, just post, and I will help you out. :)