Question #112246
Using the method of undetermined coefficients find the general solution of the differential equation
y^iv-2y"'+2y"=3e^-x+3e^-x+e^-xsinx
1
Expert's answer
2020-04-28T17:31:55-0400

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

yiv2y′′′+2y′′=3ex+2exx+exsinxy iv −2y ′′′ +2y ′′ =3e ^ {−x} +2e ^ {−x} x+e ^ {−x} sinx

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)ex+exsinxf(x)=(3+2x)e^ {−x} +e^ {−x} sinx


Therefore, a partial solution of the equation have a form

yp(x)=(A+Bx)ex+ex(Csinx+Dcosx)y_ p ​ (x)=(A+Bx)e ^ {−x} +e ^ {−x} (Csinx+Dcosx)

Find the derivatives.

yp′​=Bex(A+Bx)exex((C+D)sinx(CD)cosx)y_ p ′ ​ =Be^ {−x} −(A+Bx)e^ {−x} −e ^ {−x} ((C+D)sinx−(C−D)cosx)

yp′′​=2Bex+(A+Bx)ex+ex(2Dsinx2Ccosx)y_ p ′′ ​ =−2Be^ {−x} +(A+Bx)e^ {−x} +e^ {−x} (2Dsinx−2Ccosx)

yp′′′​=3Bex(A+Bx)ex+ex((2C2D)sinx+(2C+2D)cosx)y_ p ′′′ ​ =3Be^ {−x} −(A+Bx)e^ {−x} +e^ {−x} ((2C−2D)sinx+(2C+2D)cosx)

ypiv=4Bex+(A+Bx)exex(4Csinx+4Dcosx)y_ p iv ​ =−4Be^ {−x} +(A+Bx)e^ {−x} −e^ {−x} (4Csinx+4Dcosx)

Substituting these into the equation, obtain

14Bex+5(A+Bx)ex+ex((8D8C)sinx(8D+8C)cosx)=3ex+2xex+exsinx−14Be ^ {−x} +5(A+Bx)e^ {−x} +e^ {−x} ((8D−8C)sinx−(8D+8C)cosx) =3e^ {−x} +2xe^ {−x} +e^ {x} sinx

Rewrite the left part.

(5A14B)ex+5Bxex+ex((8D8C)sinx(8D+8C)cosx)=3ex+2xex+exsinx(5A−14B)e ^ {−x} +5Bxe ^ {−x} +e^ {−x} ((8D−8C)sinx−(8D+8C)cosx) =3e^ {−x} +2xe^ {−x} +e^ {x} sinx

Therefore,

5A14B=35A−14B=3

5B=25B=2

8D8C=18D−8C=1

8D8C=0−8D−8C=0

Solving these linear system, get that A=4325,B=2/5,C=1/16A= {43 \above{2pt} 25} ​ ,B= 2/5 ​ ,C=−1/ 16 ​ and D=1/16D=1/16

Thus, the partial solution is

yp(x)=(4325+25x)ex+ex(116cosx116sinx)y_ p ​ (x)=( {43 \above{2pt} 25} ​ + {2 \above{2pt} 5} ​ x)e^ {−x} +e^ {−x} ({1 \above{2pt} 16} ​ cosx− {1 \above{2pt} 16} ​ sinx)

Now, determine a complementary solution. Solve the characteristic equation.

λ42λ3+2λ2=0λ^ 4 −2λ^ 3 +2λ ^ 2 =0

λ2(λ22λ+2)=0λ^ 2 (λ^ 2 −2λ+2)=0


This equation have the roots λ=0λ=0 (multiplicity 2) and λ=1±iλ=1±i .Therefore, the complementary solution is

yc(x)=C1+C2x+C3excosx+C4exsinxy c ​ (x)=C_ 1 ​ +C _ 2 ​ x+C _ 3 ​ e ^ {x} cosx+C_ 4 ​ e ^ {x} sinx


So, the general solution is

y(x)=y(x)= C1+C2x+C3excosx+C4exsinxC_ 1 ​ +C _ 2 ​ x+C _ 3 ​ e ^ {x} cosx+C_ 4 ​ e ^ {x} sinx +(4325+25x)ex+ex(116cosx116sinx)( {43 \above{2pt} 25} ​ + {2 \above{2pt} 5} ​ x)e^ {−x} +e^ {−x} ({1 \above{2pt} 16} ​ cosx− {1 \above{2pt} 16} ​ sinx)


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