Question

Find the general solution of the following second order differential equation

Answer 1

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

Answer 2

you need to first solve the homogenous equation

Answer 3

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

Answer 4

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

Answer 5

@perl

Answer 6

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

Answer 7

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)

Answer 8

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

Answer 9

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