Solving a Simple Differential Equation: y''+y=5cos(x)

[physicsnewb7]

Homework Statement

y''+y=5cos(x)

Homework Equations

y=Acos(x)+Bsin(x)
e^ix=cos(x)+isin(x)

The Attempt at a Solution

I tried using compbinations of these equations to try and solve but I'm lost. Any suggestions would be greatly appreciated!

 

[fzero]

y=Acos(x)+Bsin(x)

is the solution to the homogenous equation. The particular solution must be linearly independent from them. Try multiplying by some power of x to generate a guess for the particular solution.

 

[physicsnewb7]

try multiplying what by a power of x?

 

[fzero]

try multiplying what by a power of x?

The solutions to the homogeneous equation. Try A x^a cos x and B x^b sin x and see if a and b can be chosen to get a solution.

 

[physicsnewb7]

is y= -2.5xcos(x) right?

 

[fzero]

Recheck your algebra, the sin term is actually the particular solution.

 

[physicsnewb7]

oh yeah i know i put cos by accident, but that is the solution right? I arrived at this solution by a long trial and error method. any way to quickly get at it in general?

 

[fzero]

oh yeah i know i put cos by accident, but that is the solution right? I arrived at this solution by a long trial and error method. any way to quickly get at it in general?

You have a trig function on the RHS, so the particular solution must be related to a trig function. The sin and cos are already used in the solutions to the homogeneous equations, so the particular solution can't just be sin or cos. Hence you try powers of x times sin and cos. It's just a matter of substituting the guess into the equation and examining powers and coefficients. You could even guess that you should try x cos x and x sin x because there are no powers of x on the RHS.

 

[physicsnewb7]

Thank you very much you have been very helpful. I appreciate it