Answer to Question #178212 in Differential Equations for Ashweta Padhan

Question #178212

Solve -y''-2y'+2y = e-2x

y(0)=1

y(infinity )=0


1
Expert's answer
2021-04-15T07:41:01-0400

y2y+2y=e2x-y''-2y'+2y=e^{-2x}

y(0)=1y(0)=1

y()=0y(\infty)=0

Solution:

y+2y2y=e2xy''+2y'-2y=-e^{-2x}

Characteristic equation:

r2+2r2=0r^2+2r-2=0 ;

D=4+42=23D=\sqrt{4+4\cdot 2}=2\sqrt3 ;

r1=1+3r_1=-1+\sqrt3 ; r2=13r_2=-1-\sqrt3 .

The general solution of the homogeneous differential equation:

y0=C1e(1+3)x+C2e(13)xy_0=C_1e^{(-1+\sqrt3)x}+C_2e^{(-1-\sqrt3)x}

Let's find a partial solution of the non-homogeneous equation in the form

yp=Ae2xy_p=Ae^{-2x}

yp=2Ae2xy_p'=-2Ae^{-2x}

yp=4Ae2xy_p''=4Ae^{-2x}

Substitute ypy_p, ypy_p', ypy_p'' into the origin differential equation:

4Ae2x4Ae2x2Ae2x=e2x4Ae^{-2x}-4Ae^{-2x}-2Ae^{-2x}=-e^{-2x}

A=12A=\frac12

The general solution of the non-homogeneous differential equation:

y=y0+yp=C1e(1+3)x+C2e(13)x+12e2xy=y_0+y_p=C_1e^{(-1+\sqrt3)x}+C_2e^{(-1-\sqrt3)x}+\frac12e^{-2x}

Let's apply conditions:

y(0)=C1+C2+12=1y(0)=C_1+C_2+\frac12=1

y()=0    y(\infty)=0\implies C1=0C_1=0 (because if C10C_1\neq0 then y()y(\infty)\to\infty )

C2=12C_2=\frac12

Answer: y=12e(13)x+12e2xy=\frac12e^{(-1-\sqrt3)x}+\frac12e^{-2x} .


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