non homegenous second order differential equation

eoins23456

solve the following differential equation
y''+3'=x^2+e^x-2
so the left is fairly handy c1e^-3x+c2=Yh
the right handside i cant seem to get the right answer
i tried ax^3+bx^2+cx-2x+de^x for Yp and differentiated twice and put it into the left hand side and got 1/9 for a -1/9 for b /17/27 for c and 1/4 for d

so c1e^-3x+c2+1/9x^3-1/9x^2-17/27x- 2x+1/4e^x

but wolframalpha give me
y(x) = -1/3 c_1 e^(-3 x)+c_2+1/108 (4 x (3 x^2-3 x-16)+27 e^x)

did i make the wrong guess for Yp?

Michael Collins

I assume you meant:

[latex] \displaystyle y'' + 3y' = e^x + x^2 - 2 [/latex]

which as you say gives you

[latex] \displaystyle y_h(x) = c_1 e^{-3x} + c_2 [/latex]

Your answer for yp is nearly correct, c should be -16/27. Don't worry about Wolfram Alpha, it can give you a simple answer in a complicated form.

You can always check your answer yourself by subbing yh and yp back into the differential equation.

When you sub in yh on the left hand side, you should get the result = 0
When you sub in yp on the left hand side, you should get the right hand side as the result.