Using the method of undermined coefficients, find the general solution of the differential equation y^iv-2y'''+2y''=3e^-x+2e^-xx+e^-x sin x
yiv−2y′′′+2y′′=3e−x+2e−xx+e−xsinx
To determine the general solution needed find a complementary and partial solutions. First, determine a partial solution.
Rewrite the right part of the equation as
f(x)=(3+2x)e−x+e−xsinx
Therefore, a partial solution of the equation have a form
yp(x)=(A+Bx)e−x+e−x(Csinx+Dcosx)
Find the derivatives.
yp′=Be−x−(A+Bx)e−x−e−x((C+D)sinx−(C−D)cosx)
yp′′=−2Be−x+(A+Bx)e−x+e−x(2Dsinx−2Ccosx)
yp′′′=3Be−x−(A+Bx)e−x+e−x((2C−2D)sinx+(2C+2D)cosx)
ypiv=−4Be−x+(A+Bx)e−x−e−x(4Csinx+4Dcosx)
Substituting these into the equation, obtain
−14Be−x+5(A+Bx)e−x+e−x((8D−8C)sinx−(8D+8C)cosx)=3e−x+2xe−x+exsinx
Rewrite the left part.
(5A−14B)e−x+5Bxe−x+e−x((8D−8C)sinx−(8D+8C)cosx)=3e−x+2xe−x+exsinx
Therefore,
5A−14B=3
5B=2
8D−8C=1
−8D−8C=0
Solving these linear system, get that A=2543,B=2/5,C=−1/16 and D=1/16
Thus, the partial solution is
yp(x)=(2543+52x)e−x+e−x(161cosx−161sinx)
Now, determine a complementary solution. Solve the characteristic equation.
λ4−2λ3+2λ2=0
λ2(λ2−2λ+2)=0
This equation have the roots λ=0 (multiplicity 2) and λ=1±i .Therefore, the complementary solution is
yc(x)=C1+C2x+C3excosx+C4exsinx
So, the general solution is
y(x)= C1+C2x+C3excosx+C4exsinx +(2543+52x)e−x+e−x(161cosx−161sinx)
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