Answer to Question #116886 in Differential Equations for Vikas

Question

using the method of undetermined coefficient find the general solution of the differential equation dy^4/dx^4-2dy^3/dx^3+2dy^2/dx^2=3/e^x+2x/e^x +sinx /e^x

Answer 1

this equation can be written as (D⁴-2D³+2D²)y = (3+2x)e^-x+sinx.e^-x ....(1)

the auxiliary equation is

D⁴-2D³+2D²=0

D= 0,0,1+i,1-i

so, the complimenatry function is

y= c₁₊c₂x + e^x(c₃ cosx + c₄sinx)

let trial solution be

z= e^-x(A sinx + B cosx) + e^-x(D+Ex)

Dz= -e^-x( A sinx + B cosx + D+Ex) + e^-x( A cosx - B sinx + E )

D²z= e^-x( D+Ex) -2e^-x(A cosx - B sinx +E)

D³z = -e^-x(-2A sinx -2B cosx + D+Ex) + 3e^-x(A cosx - B sinx +E)

D⁴z = e^-x( -5A sinx -5 B cosx + D+Ex) - e^-x(A cosx - B sinx + 4E )

putting these values in (1), we get

e^-x[ sinx(-9A+9B) + x (5E) + 5D -14E + cosx(-9B-9A)] = e^-x(3+2x+ sinx)

comparing coefficients

A=-1/18

B=1/18

D=43/25

E= 2/5

So, the solution is

y= c₁+(c₂+0.4e^-x)x + e^x(c₃ cosx + c₄sinx) + e^-x[( -1/18) . sinx + (1/18) . cosx + (43/25 )]