Answer to Question #122585 in Differential Equations for jse

Question

1. Find a homogeneous linear differential equation with constant coefficients whose general solution is given.

y = c1e−x cos x + c2e−x sin x

A. y'' + 2y' + 2y = 0
B. y'' − 2y' + 2y = 0
C. y''' + 2y'' + 2y' = 0
D. y''' − 2y'' + 2y' = 0
E. y'' + 1 = 0

2. Find the general solution of the given second-order differential equation.
y'' + 36y = 0

y(x) = ____

Answer 1

1.

Solution:

y = c₁e^-xcos x + c₂^-x sin x,

k₁=-1-i, k₂=-1+i,

(k-k₁)(k-k₂)=(k+1+i)(k+1-i)=k²+2k+2,

y''+2y'+2y=0

Answer:

A.y''+2y'+2y=0.

2.

Solution:

y''+36y=0,

k²+36=0,

k²=-36,

k₁=-6i, k₂=6i,

y(x)=c₁cos(6x)+c₂sin(6x).

Answer:

y(x)=c₁cos(6x)+c₂sin(6x).