Use Eigenvalues and Eigenvectors to find the general solution of the following system of differential equations:

Question:

 

Use Eigenvalues and Eigenvectors to find the general solution of the following system of differential equations:

y1' = -4y1 - 5y2

y2' = 5y1 + 6y2

Solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Use variation of parameters to find the general solution to the following differential equation. x^2y'' - 3xy' + 4y = x^2 ln(x)

Question.

Use variation of parameters to find the general solution to the following differential equation.

x²y'' - 3xy' + 4y = x² ln(x)

Hint: The solution to the homogeneous problem is y_h = c₁x² + c₂x²ln(x)

Solution:

Solve the following differential equation using Laplace Transforms. y'' + y' - 2y = -4 , y(0) = 0 , y'(0) = 0

Question.

Solve the following differential equation using Laplace Transforms.

y'' + y' - 2y = -4 , y(0) = 0 , y'(0) = 0

Solution:

Find the fundamental matrix Ψ(t) satisfying Ψ(0) = I for the system of equations. X'(t) = [-1 -4 ; 1 -1] X(t)

Question.

Find the fundamental matrix Ψ(t) satisfying Ψ(0) = I for the system of equations.

Solution:

Solve the following differential Equation using undetermined coefficients. y'' - 15y' + 57y = 9e^(8t)

Solve the following differential Equation using undetermined coefficients.

y'' - 15y' + 57y = 9e^8t

Solution: