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:

Solve differential equation using Laplace Transforms. y'' - 6y' + 5y = te^t , y(0) = 2 , y'(0) = -1

Question.


Solve differential equation using Laplace Transforms.

y'' - 6y' + 5y = te^t , y(0) = 2 , y'(0) = -1

Solution:

Solve by using Laplace Transforms. y'''' - 9y = 0 , y(0) = 1, y'(0) = 0 , y''(0) = 0 , y'''(0) = 1

Question. Solve by using Laplace Transforms.
y'''' - 9y = 0 , y(0) = 1, y'(0) = 0 , y''(0) = 0 , y'''(0) = 1

Solution:

Use Eigenvalues and Eigenvectors to find the general solution of the following system of differential equations: y1' = 4y1 + 5y2 ; y2' = -2y1 + 6y2

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

y1' = 4y1 + 5y2
y2' = -2y1 + 6y2

Solution:

Write the form of particular Solution with the method of undetermined coefficients.

Write the form of particular Solution with the method of undetermined coefficients.

A) y'' + 6y' + 9y = 5cos(2x)

B) y'' + 10y' + 24y = 3xe^6x

C) y'' - 9y' + 18y = 3e2x + 7x³sin(4x)

Solution:

Convert the initial value problem into a first order system. Give the matrix A which determines the system and the initial vector x(0). y'' - 3y' - 10y = 3e^(-2t) + 2e^(5t) y(0)=2, y'(0)= -3

Convert the initial value problem into a first order system. Give the matrix A which determines the system and the initial vector x(0).

y'' - 3y' - 10y = 3e^-2t + 2e^5t

y(0)=2, y'(0)= -3

Solution:

Find the general solution of the following equation. y'' - 4y' + 4y = (e^2x)/x

Find the general solution of the following equation.

y'' - 4y' + 4y = (e^2x)/x

Solution:

Find the inverse Laplace transform 6/(s^3+3s)

Question.

Find the inverse Laplace transform 6/(s³+3s)

Solution: