Determine a homogeneous linear differential equation with constant coefficients having the following as two of its solutions: Solutions: x sin(3x) and x^2 + 8

Question.

Determine a homogeneous linear differential equation with constant coefficients having the following as two of its solutions:

Solutions: x sin(3x) and x² + 8


Solution:

Solve the following differential equation using annihilator method y'' + 3y' -2y = e^(5t) + e^t

Question.


Solve the following differential equation using annihilator method

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

Solution:

Solve the following differential equation by using the method of undetermined coefficients. y’’ + 5y’ – 6y = e^t , y(0) = 1 , y’(0) = 0

Question.

Solve the following differential equation by using the method of undetermined coefficients.

y’’ + 5y’ – 6y = e^t , y(0) = 1 , y’(0) = 0

Solution:

Solve the differential equation. y’’ –(y’)^2014 = 0

Question.

Solve the differential equation.

y’’ –(y’)²⁰¹⁴ = 0

Solution:

Solve the IVP. y’ – (1/t)y = e^t y^2 , y(1) = 305

Question.

Solve the IVP.

y’ – (1/t)y = e^ty² , y(1) = 305

Solution:

Find the general Solution: dy/dx = y/x (1 – y/x)

Question.

Find the general Solution:

dy/dx = y/x (1 – y/x)

Solution:

Solve the exact differential equation. (e^y + x^2 + sin(y)) dx + x(e^y + cos(y)) dy = 0

Question.

Solve the exact differential equation.

(e^y + x² + sin(y)) dx + x(e^y + cos(y)) dy = 0

Solution:

Find the general solution of y'' + y' - 6y = 0 and take the limit of y as t approaches positive infinity.

Question.


Find the general solution of y'' + y' - 6y = 0 and take the limit of y as t approaches positive infinity.



Solution:

Use Euler's method to obtain a four-decimal approximation of the indicated value. Use h = 0.5 ; y' = 3x^2/2y , y(0) = 1, y(2)

Question.

Use Euler's method to obtain a four-decimal approximation of the indicated value. Use h = 0.5

y' = 3x²/2y , y(0) = 1, y(2)


Solution:

Solve the system of differential equations by systematic elimination. dx/dt = -2x - y ; dy/dt = -4y

Question.

Solve the system of differential equations by systematic elimination.

dx/dt = -2x - y
dy/dt = -4y

Solution:

Use Euler's method to obtain a four-decimal approximation of the indicated value. use h = 0.5 y' = 3x^2/2y ; y(0) = 1 ; y = 2

Question.

Use Euler's method to obtain a four-decimal approximation of the indicated value. use h = 0.5

y' = 3x²/2y ; y(0) = 1 ; y = 2

Solution:

Solve the following Cauchy-Euler equation x^2y'' + 5xy' - 5y = 0

Question.

Solve the following Cauchy-Euler equation x²y'' + 5xy' - 5y = 0

Solve the IVP (2xy + x^4)dx + (x^2 + y^2)dy =0 y(0)=1

Question.

Solve the IVP

(2xy + x⁴)dx + (x² + y²)dy =0

y(0)=1

Solve the IVP. dy/dt - 2ty = -6t^2 e^(t^2) ; y(0) = 1

Question.

Solve the IVP.
dy/dt - 2ty = -6t^2 e^(t^2) ; y(0) = 1

Solution:

For the below ordinary differential equation, state the order and determine if the equation is linear or nonlinear. Then find the general solution of the ordinary differential equation. Verify your solution. dy/dx = 2y

Question.

For the below ordinary differential equation, state the order and determine if the equation is linear or nonlinear. Then find the general solution of the ordinary differential equation. Verify your solution.

dy/dx = 2y


Solution:

Find the differential equation that has y^2 = Cx + 3 as its general solution

Question.

Find the differential equation that has y² = Cx + 3 as its general solution

Solution:

Solve the differential equation y' = 8x - y

Question.

Solve the differential equation
y' = 8x - y

Solution:

Find inverse Laplace transform of e^(-2s)/( s^2(s-1) )

Question.

Find inverse Laplace transform of e^-2s/( s²(s-1) )

Solution:

Use Euler’s method with h = 0.1 and h = 0.05 to calculate y(1.2) ; dy/dx = 2x – 3y + 1 , y(1) = 5

Question.

Use Euler’s method with h = 0.1 and h = 0.05 to calculate y(1.2)

dy/dx = 2x – 3y + 1 , y(1) = 5

Solution:

Find the Laplace Transform of f(t) = {cos t , 0 <= t <= Pi ; 0, t >= Pi} using definition of Laplace Transform.

Question.

Find the Laplace Transform of f(t) = {cos t , 0 <= t <= Pi ; 0, t >= Pi} using definition of Laplace Transform.

Solution:

Show that following differential equation is not exact. (3x^2y^4 + 2xy)dx + (2x^3y^3 - x^2)dy = 0 Then find an integrating factor to solve the differential equation.

Question.

Show that following differential equation is not exact.

(3x²y⁴ + 2xy)dx + (2x³y³ - x²)dy = 0

Then find an integrating factor to solve the differential equation.

Solution:

Solve the initial value problem. y’’ – 5y’ + 6y = 0 , y(0) = 3 , y’(0) = 5

Question.

Solve the initial value problem.

y’’ – 5y’ + 6y = 0 , y(0) = 3 , y’(0) = 5

Solution:

Use Euler's Method to compute y(4) given that y' = y - x^2 , y(1)

Question.

Consider the initial value problem y'= y - x², y(1) = 2.

(a) For this IVP, write down Euler's method for y_n+1 in terms of y_n for a general stepsize h.

(b) Using one step of Euler's method approximate y(4).

(c) Using three steps of Euler's method approximate y(4).

Solution:

Solve the differential equation dP/dt = kP - C

Question.

Solve the differential equation dP/dt = kP - C

Solution:

Solve the separable differential equation dy/dx = -0.8/cos(y) with initial condition y(0) = pi/4

Question.

Solve the separable differential equation dy/dx = -0.8/cos(y) with initial condition y(0) = pi/4

Solution:

Solve the separable differential equation y’ = sqrt(-2y + 41) with initial condition y(-5) = 8

Question.

Solve the separable differential equation y’ = sqrt(-2y + 41) with initial condition y(-5) = 8

Solution:

Use Euler’s method to calculate y(0.5) using h = 0.05 and h = 0.1 y’ = e^(-y) , y(0) =0

Question.

Use Euler’s method to calculate y(0.5) using h = 0.05 and h = 0.1

y’ = e^(-y) , y(0) =0

Solution:

Solve the following differential equation using reduction of order. t^2 y’’ + 9ty’ -3 = 0 Also, solve the initial value problem subject to the initial conditions y(1) = 4 and y’(1) = 3/8 , t > 0

Question.

Solve the following differential equation using reduction of order.

t²y’’ + 9ty’ -3 = 0

Also, solve the initial value problem subject to the initial conditions y(1) = 4 and y’(1) = 3/8 , t > 0

Solution:

Solve the differential Equation dy/dt = yt^2 + 4

Question.

Solve the differential Equation

dy/dt = yt² + 4

Solution:

Use Euler’s method with step size 0.5 to compute the approximate y-values y(1.5), y(2), y(2.5) and y(3)

Question.

Use Euler’s method with step size 0.5 to compute the approximate y-values y(1.5), y(2), y(2.5) and y(3)

Solution:

Solve the following exact differential equation. ( 6xye^(2y) + 4x^3 ) dx + ( 3x^2 e^(2y) + 6x^2 ye^(2y) - 2sin(2y) ) dy = 0

Question.

Solve the following exact differential equation.

( 6xye^2y + 4x³ ) dx + ( 3x²e^2y + 6x²ye^2y - 2sin(2y) ) dy = 0

Solution:

Check if the following differential equation is exact? If so, solve. (4e^x sin(y) - 2y)dx + (-2x + 4e^x cos(y))dy = 0

Question.

Check if the following differential equation is exact? If so, solve.

(4e^x sin(y) - 2y)dx + (-2x + 4e^x cos(y))dy = 0

Solution:

dy/dx = (1+y^2)/(y+yx^2)

Question.

dy/dx=(1+y^2)/(y+yx²)


Solution:

Determine the general solution of the first order differential equation: dy/dx = 2y ln y + xye^(2x)cos(3x) , y>0

Question.

Determine the general solution of the first order differential equation:
dy/dx = 2y ln y + xye^(2x)cos(3x) , y>0

Solution:

Solve the IVP y' + y = sin(t) + cos(t) with y(0) = 1 by A) method of undetermined coefficients and B) method of variation of parameters

Question.

Solve the IVP y' + y = sin(t) + cos(t) with y(0) = 1 by

A) method of undetermined coefficients and

B) method of variation of parameters


Solution:

Solve the IVP y' + y = te^(-t) with y(0) = 0 by A) method of undetermined coefficients and B) method of variation of parameters

Question.

Solve the IVP y' + y = te^(-t) with y(0) = 0 by

A) method of undetermined coefficients and

B) method of variation of parameters


Solution:

Solve the IVP y' + y = (e^t) cos(t) + (e^t) sin(t) with y(0)=1 by A) method of undetermined coefficients B) method of variation of parameters

Question.

Solve the IVP y' + y = (e^t) cos(t) + (e^t) sin(t) with y(0)=1 by

A) method of undetermined coefficients

B) method of variation of parameters

Solution:

Solve the IVP xy'-2y=x^2+6 , with y(1)=1 by method of variation of parameters

Question.

Solve the IVP xy' - 2y = x² + 6 , with y(1)=1 by method of variation of parameters

Solution:

Solve the IVP y' + (tan t)y = 2sec(t) tan(t), y(0)=0 by method of variation of parameters Plot the solution y=y(t) on 0<= t < (pi/2) and compute lim as t -> (pi/2^-) of y(t)

Question.

Solve the IVP y' + (tan t)y = 2sec(t) tan(t), y(0)=0 by method of variation of parameters

Plot the solution y=y(t) on 0<= t < (pi/2) and compute lim as t -> (pi/2^-) of y(t)

Solution:

Use Euler's method to obtain a four-decimal approximation of the indicated value. y' = 2x - 3y + 1, y(1) = 3; y(1.2)

Question.

Use Euler's method to obtain a four-decimal approximation of the indicated value. 

y' = 2x - 3y + 1, y(1) = 3;   y(1.2)

Solution:

Solve the given Linear DE subject to the indicated initial condition x dy/dx + y = x cos(x) ; y(pi) = 1

Question.


Solve the given Linear DE subject to the indicated initial

x dy/dx + y = x cos(x) ; y(pi) = 1


Solution: