Answer to Question #195875 in Differential Equations for Prince

Question

 Solve the equation of the Linear Homogeneous Differential Equations with Constant 

Coefficients.

1. Solve y’’ – y’ – 2y = 0

2. Solve y’’ + 4y = 0

3. (D3 + 3D2 – 4D)y = 0

4. (D2 – 6D + 9)y = 0

Answer 1

1) $y''-y'-2y=0$

Auxiliary equation is-

$m^2-m-2=0\\m^2-2m+m-2=0\\m(m-2)+1(m-2)=0\\(m-2)(m+1)=0\\m=2,-1$

Solution is-

$y=c_1e^{2x}+c_2e^{-x}$

$2)y''+4y=0$

Auxiliary equation is-

$m^2+4=0\\(m+2i)(m-2i)=0\\m=\pm2i$

Solution is-

$y=c_1 cos2x+c_2sin2x$

$(3)(D^3+3D^2-4D)y=0$

Auxiliary equation is-

$m^3+3m^2-4m=0\\m(m^2+3m-4)=0\\m(m^2+4m-m-4)=0\\m(m+4)(m-1)=0\\m=0,1,-4$

Solution is-

$y=c_1+c_2e^x+c_3e^{-4x}$

$(4) (D^2 – 6D + 9)y = 0$

Auxiliary equation-

$m^2-6m+9=0\\(m-3)^2=0\\m=3,3$

Solution is-

$y=(c_1+c_2x)e^{3x}$