Find the general solution of the following second order differential equation

Question

anonymous · Answer

$$\frac{ d ^{2}y }{ dx ^{2} } + \frac{ dy }{ dx } = x ^{2}$$

perl · Answer

you need to first solve the homogenous equation

perl · Answer

The general solution is:
 
   y(x)  =  y_c(x)+y_p(x)  =  x^3/3-x^2+2 x+c_1 e^(-x)+c_2

anonymous · Answer

how do you solve the homogenous equation? the x^2 is what's putting me off, if it was x i would easily be able to solve it

anonymous · Answer

@perl

perl · Answer

to solve the homogenous, you have 
y ' ' + y ' = 0

perl · Answer

the characteristic equation is 
r^2 + r = 0 

r ( r + 1) = 0

r = 0, r = -1

y = c_1 * e^ (0*x) + c2 * e^(-1*x)

perl · Answer

that simplifies to 
y = c_1 * 1 + c2 * e^ -x

anonymous · Answer

thanks so much @perl it all makes sense now! just having one of those days haha