Solving a 2nd Order Differential Equation with Auxiliary Method

[angel]

ok
im trying to solve the following equation using standard aux method:

d^2y/dx^2 + 3dy/dy +2y = cos x with conditions x(0)=-3 and x'(0) = 3

my aux eqn is:

ae^x + be^-2x

and my yp is;

a sin kx + b cos kx

i differentiate this twice and substitute into the original equation, and i get:

(-a sin x - b cos x)+3(a cos x-b sin x) + 2(a sin x+ b cos x) = cos x.

Now at this stage I am not sure if this is right or not, can someone please confirm if this is right or not.

I ve done this question using the laplace method and the answer i get is;

-19/2e^x +32/5 e^2x + 1/10 cos x - 3/10 sin x.

Here my aux equation is not the same as what i get when i apply the laplace method.

Im not sure what is going wrong.
Someone help me please.

 

[matt grime]

you have a and b representing two different numbers simlutaneously.

if you're not sure which of e^2x or e^-2x is correct why don't you subs it into the equation to see which one works? i think it is -2, i also think you want e^-x and not e^x since the aux equation is m^2+3m+2=(m+1)(m+2)

 

[Max0526]

solution

Hi;
Here is your solution:
$$y(x)=\frac{3sinx}{10}+\frac{cosx}{10}-\frac{7e^{-x}}{2}+\frac{2e^{-2x}}{5}$$
Best of luck,
Max.
P.S. Obtained by Maxima: http://maxima.sourceforge.net/download.shtml