Differential Equations Answers

Solve the differential equation by variation of parameters, subject to the initial conditions

y(0) = 1, y'(0) = 0.


9y'' − y = x(e^x/3)


y(x) = ____

The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation.


y'' − 3y' + 2y = 7e^3x; y1 = e^x


y1(x) = ____

yp(x) = _____

1. The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2,


y2 = y1(x) ∫ e^−∫P(x) dx / y upper 2 and lower 1 (x) dx


as instructed, to find a second solution y2(x).

y'' − 4y' + 4y = 0; y1 = e^2x


y2 = _____

Verify that y1(x) = x is a solution of xy'' − xy' + y = 0. Use reduction of order to find a second solution y2(x) in the form of an infinite series.


A. y2 = x ln x + x^2 + x^3/ 2·2! + x^4/ 3·3! + x^5/4·4! +...

B. y2 = −1 + x ln x + x^2/ 2 + x^3/ 2·3! + x^4/ 3·4! + ...

C. y2 = ln x + x + x^2/ 2·2! + x^3/ 3·3! + x^4/ 4·4! + ...

D. y2 = ln x + x + x^2/ 2! + x^3/ 3! + x^4/ 4! + ...

E. y2 = − 1/ x + ln x + x/2 + x^2/ 2·3! + x^3/ 3·4! + ...


Conjecture an interval of definition for y2(x).

A. [0, ∞)

B. [−1, 1]

C. (−1, 1)

D. [−1, ∞)

E. (0, ∞)

The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2,


y2 = y1(x) ∫ e^−∫P(x) dx / y upper 2 and lower 1 (x)


as instructed, to find a second solution y2(x).


x^2 y'' − xy' + 17y = 0; y1 = x sin(4 ln x)


y2 = ____

The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution of the given nonhomogeneous equation.


y'' − 25y = 2; y1 = e^−5x


y1(x) = _____

y2(x) = ______

1. Solve the given differential equation by undetermined coefficients.

y'' − 8y' + 16y = 12x + 6


y(x) =_____


2. Solve the given boundary-value problem.

y'' + 7y = 7x, y(0) = 0, y(1) + y'(1) = 0


y(x) = _____

1. Solve the given differential equation by undetermined coefficients.

y'' + 3y = −48x^2 (e^3x).


y(x) =____



2. Find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra.


y^(4) + 2y'' + y = 1 cos x − 3x sin x


yp = _____

1. Solve the given differential equation by undetermined coefficients.

y'' + 6y' + 9y = −xe^4x


y(x) =____


2. Solve the given differential equation by undetermined coefficients.

y''' − 3y'' + 3y' − y = e^x − x + 21


y(x)= _____

1. Solve the given differential equation by undetermined coefficients.

y'' − 2y' + 5y = e^x sin x


y(x) =____


2. Solve the given differential equation by undetermined coefficients.

y'' − y' − 2y = e^2x


y(x) =___