Differential Equations

Your highest order differencial is 4 so chances are youll want something to plug into that part of the equ

Have no clue what you're asking for. Can you provide a more clear example?

>>9000374
Ifs its to the n order, then you take the n derivative.

If it has n constants then you take it to the n derivative, then solve

Integrate both sides of the equation with respect to x twice; then use a trial function y (x)=ce^(rx). You'll get something like:

(r^2+2r+2)c = 1/4(x^4)×e^(-rx)

Solve for r:

(r^2+2r+2)c=((r+1)^2+1)c=(1/4)x^4×e^(-rx)

r=-1+-(((1/4c)x^4×e^(-rx))-1)^(1/2)

input and you have your general equation.

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Whoops, the actual general equation is:

r=-1+-((((1/4c3)x^4+c1+c2)×e^(-rx))-1)^(1/2)

Since 3x^2 was originally integrated twice.

>>9000374
How about fucking practicing

And to your final question: the characteristic equation you want to use will be governed by what trial function you input. For instance, if you have y''''+y'=0, you can input the trial function y (x)=e^rx to get r^4+r=0 or you can integrate both sides once to obtain y'''+y=0 which produces r^3+1=0.

Ultimately, the characteristic function you want to use will correspond to the highest order differential in the initial equation.

>>9002182
Sometimes a bit of guidance goes farther than hours of practice.

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Help!

x^2*y''+sinx*y=0

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First,

X^2×y''=-sinx y, isolate functions involving x on one side, y on the other.

(1/y)y''=-(1/x^2)sinx

Use y=ce^(rx), then: r^2=(1/x^2)sinx which implies that r= +or-((1/x^2)sinx)^1/2.

Then, y=c1exp ((1/x^2)sinx)+c2exp (-(1/x^2)sinx).

Help me solve y''' -2cy'+bx^2=g (x) where g (x) is arbitrary

>>9004290
Fourier transform: y=int-inf,inf f (w)e^iwx dw

(-iw^3 + 2ciw+bw^2)f (w)=g

Solve for f (w), choose g (w), and integrate over w to get y (x)

>>9000374
You need to prevent duplication in your particular solution, you should see right off the bat that your x^2 will have the 3rd and 4th derivative be zero

You pretty much answer your own question, because it's 4th order you want your guess to include ax^4
It's pretty much ALWAYS going to follow that pattern where your highest order should match your highest x power.
It's possible that you'll find some of your constants to be zero but it's better to find a constant to be zero than to keep starting over.