Answer to Question #122592 in Differential Equations for jse

Question

1. Find linearly independent functions that are annihilated by the given differential operator. (Give as many functions as possible. Use x as the independent variable. Enter your answers as a comma-separated list.)

D4

2. Solve the given differential equation by undetermined coefficients.
y'' + y' + y = x sin x

y(x) = ______

Answer 1

Solution:

1.D⁴y=0

y⁽⁴⁾=0

$\lambda$ ⁴=0

$\lambda$ ₁=0, k=4

y₁(x)=1,y₂(x)=x,y₃(x)=x²,y₄(x)=x³.

Answer: 1, x, x², x³.

Solution:

y'' + y' + y = x sin x

y'' + y' + y = 0

$\lambda$ ²+$\lambda$ +1=0

$\lambda$ ₁=-1/2-i3^1/2/2, λ ₂=-1/2+i3^1/2/2.

y₀(x)=C₁e^-x/2cos(3^1/2x/2)+C₂e^-x/2sin(3^1/2x/2),

y_p(x)=(Ax+B)sin(x)+(Cx+D)cos(x)

y_p'(x)=(-Cx+A-D)sin(x)+(Ax+B+C)cos(x)

yp''(x)=(-Ax-B-2C)sin(x)+(-Cx+2A-D)cos(x)

xsin(x):A-C-A=1,

xcos(x):C+A-C=0,

sin(x):B+A-D-2C-B=0,

cos(x):D+B+C+2A-D=0.

=>C=-1,A=0,D=2,B=1.

y_p(x)=sin(x)+(-x+2)cos(x),

y(x)=y₀(x)+y_p(x)

Answer:

y(x)=C₁e^-x/2cos(3^1/2x/2)+C₂e^-x/2sin(3^1/2x/2)+sin(x)+(-x+2)cos(x)