# Thread: solve given a yh1

1. ## solve given a yh1

Hello,
I cannot for the life of me remember how to do this. Can someone help please?

Solve the following differential equation xy''+y'=12x^2 given that yh1 = 1 is a solution of the homogeneous equation.

Thanks.

2. The procedure to arrive, once you know a solution of this type of equation, to a second solution independent from it, is illustrated here...

http://www.mathhelpforum.com/math-help/calculus/82519-variation-parameter-2nd-order-ode.html

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Normally this could be handled using reduction of order where we would make the substitution:

y(x) = yh*z(x)

However since yh = 1 we don't need to do this

Let z = y ' directly and we have

xz' + z =12x^2

(xz) ' =12x^2

xz = 4x^3 + C1

z = 4x^2 +C1/x Note 1/x is the other homogeneous solution

now solve y ' = 4x^2 +C1/x