Exam 17: Second-Order Differential Equations

The solution of the initial-value problem $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is called a Bessel function of order 0. Solve the initial - value problem to find a power series expansion for the Bessel function.

(Multiple Choice)

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Question 41

Use power series to solve the differential equation. $y ^ { \prime \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { \prime } ( 0 ) = 0$

(Multiple Choice)

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Question 42

Solve the differential equation. $3 y ^ { \prime \prime } + y ^ { \prime } = 0$

(Short Answer)

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Question 43

Solve the initial-value problem. $y ^ { \prime \prime } + 8 y ^ { \prime } + 41 y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 7$

(Multiple Choice)

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Question 44

Use power series to solve the differential equation. $y ^ { \prime } = 7 x ^ { 3 } y$

(Multiple Choice)

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Question 45

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - y ^ { \prime \prime } = e ^ { 2 x }$

(Multiple Choice)

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Question 46

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + 25 y = x$

(Short Answer)

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Question 47

Graph the particular solution and several other solutions. $2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x$

(Multiple Choice)

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Question 48

Solve the differential equation. $4 y ^ { \prime \prime } + y = 0$

(Multiple Choice)

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Question 49

A spring with a mass of 2 kg has damping constant 14, and a force of $4.8$ N is required to keep the spring stretched $0.4$ m beyond its natural length. Find the mass that would produce critical damping.

(Short Answer)

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Question 50

Use power series to solve the differential equation.. $\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0$

(Multiple Choice)

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Question 51

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + y = \sec x , \frac { \pi } { 4 } < x < \frac { \pi } { 2 }$

(Multiple Choice)

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Question 52

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } - y ^ { \prime } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0$

(Short Answer)

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Question 53

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } - 3 y ^ { \prime } = \sin 9 x$

(Multiple Choice)

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Question 54

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 7 e ^ { - x }$

(Short Answer)

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Question 55

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 5 y ^ { \prime } - 36 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

(Multiple Choice)

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Question 56

Solve the differential equation. $y ^ { \prime \prime } - 8 y ^ { \prime } + 25 y = 0$

(Multiple Choice)

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Question 57

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 3 y ^ { \prime } = e ^ { 3 x }$

(Short Answer)

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Question 58

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } + 5 y ^ { \prime } + 6 y = x ^ { 2 }$

(Multiple Choice)

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Question 59

A spring has a mass of $1$ kg and its damping constant is $c = 10$ . The spring starts from its equilibrium position with a velocity of $1$ m/s. Graph the position function for the spring constant $k = 20$ .

(Multiple Choice)

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Question 60

The solution of the initial-value problem $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is called a Bessel function of order 0. Solve the initial - value problem to find a power series expansion for the Bessel function.

Use power series to solve the differential equation. $y ^ { \prime \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { \prime } ( 0 ) = 0$

Solve the differential equation. $3 y ^ { \prime \prime } + y ^ { \prime } = 0$

Solve the initial-value problem. $y ^ { \prime \prime } + 8 y ^ { \prime } + 41 y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 7$

Use power series to solve the differential equation. $y ^ { \prime } = 7 x ^ { 3 } y$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - y ^ { \prime \prime } = e ^ { 2 x }$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + 25 y = x$

Graph the particular solution and several other solutions. $2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x$

Solve the differential equation. $4 y ^ { \prime \prime } + y = 0$

A spring with a mass of 2 kg has damping constant 14, and a force of $4.8$ N is required to keep the spring stretched $0.4$ m beyond its natural length. Find the mass that would produce critical damping.

Use power series to solve the differential equation.. $\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + y = \sec x , \frac { \pi } { 4 } < x < \frac { \pi } { 2 }$

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } - y ^ { \prime } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } - 3 y ^ { \prime } = \sin 9 x$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 7 e ^ { - x }$

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 5 y ^ { \prime } - 36 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

Solve the differential equation. $y ^ { \prime \prime } - 8 y ^ { \prime } + 25 y = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 3 y ^ { \prime } = e ^ { 3 x }$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } + 5 y ^ { \prime } + 6 y = x ^ { 2 }$

A spring has a mass of $1$ kg and its damping constant is $c = 10$ . The spring starts from its equilibrium position with a velocity of $1$ m/s. Graph the position function for the spring constant $k = 20$ .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Final Exam

Filters

Exam 17: Second-Order Differential Equations

Use power series to solve the differential equation. y′′+x2y=0,y(0)=6,y′(0)=0y ^ { \prime \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { \prime } ( 0 ) = 0y′′+x2y=0,y(0)=6,y′(0)=0

Solve the differential equation. 3y′′+y′=03 y ^ { \prime \prime } + y ^ { \prime } = 03y′′+y′=0

Solve the initial-value problem. y′′+8y′+41y=0,y(0)=1,y′(0)=7y ^ { \prime \prime } + 8 y ^ { \prime } + 41 y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 7y′′+8y′+41y=0,y(0)=1,y′(0)=7

Use power series to solve the differential equation. y′=7x3yy ^ { \prime } = 7 x ^ { 3 } yy′=7x3y

Solve the differential equation using the method of variation of parameters. y′′−y′′=e2xy ^ { \prime \prime } - y ^ { \prime \prime } = e ^ { 2 x }y′′−y′′=e2x

Solve the differential equation using the method of variation of parameters. y′′+25y=xy ^ { \prime \prime } + 25 y = xy′′+25y=x

Graph the particular solution and several other solutions. 2y′′+3y′+y=2+cos⁡2x2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x2y′′+3y′+y=2+cos2x

Solve the differential equation. 4y′′+y=04 y ^ { \prime \prime } + y = 04y′′+y=0

A spring with a mass of 2 kg has damping constant 14, and a force of 4.84.84.8 N is required to keep the spring stretched 0.40.40.4 m beyond its natural length. Find the mass that would produce critical damping.

Use power series to solve the differential equation.. (x2+1)y′′+xy′−y=0\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0(x2+1)y′′+xy′−y=0

Solve the differential equation using the method of variation of parameters. y′′+y=sec⁡x,π4<x<π2y ^ { \prime \prime } + y = \sec x , \frac { \pi } { 4 } < x < \frac { \pi } { 2 }y′′+y=secx,4π​<x<2π​

Solve the boundary-value problem, if possible. y′′−y′−2y=0,y(−1)=1,y(1)=0y ^ { \prime \prime } - y ^ { \prime } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0y′′−y′−2y=0,y(−1)=1,y(1)=0

Solve the differential equation using the method of undetermined coefficients. y′′−3y′=sin⁡9xy ^ { \prime \prime } - 3 y ^ { \prime } = \sin 9 xy′′−3y′=sin9x

Solve the differential equation using the method of undetermined coefficients. y′′−4y′+5y=7e−xy ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 7 e ^ { - x }y′′−4y′+5y=7e−x

Solve the boundary-value problem, if possible. y′′+5y′−36y=0,y(0)=0,y(2)=1y ^ { \prime \prime } + 5 y ^ { \prime } - 36 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1y′′+5y′−36y=0,y(0)=0,y(2)=1

Solve the differential equation. y′′−8y′+25y=0y ^ { \prime \prime } - 8 y ^ { \prime } + 25 y = 0y′′−8y′+25y=0

Solve the differential equation using the method of variation of parameters. y′′−3y′=e3xy ^ { \prime \prime } - 3 y ^ { \prime } = e ^ { 3 x }y′′−3y′=e3x

Solve the differential equation using the method of undetermined coefficients. y′′+5y′+6y=x2y ^ { \prime \prime } + 5 y ^ { \prime } + 6 y = x ^ { 2 }y′′+5y′+6y=x2

A spring has a mass of 111 kg and its damping constant is c=10c = 10c=10 . The spring starts from its equilibrium position with a velocity of 111 m/s. Graph the position function for the spring constant k=20k = 20k=20 .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Final Exam

Filters

Use power series to solve the differential equation. $y ^ { \prime \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 6 , y ^ { \prime } ( 0 ) = 0$

Solve the differential equation. $3 y ^ { \prime \prime } + y ^ { \prime } = 0$

Solve the initial-value problem. $y ^ { \prime \prime } + 8 y ^ { \prime } + 41 y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 7$

Use power series to solve the differential equation. $y ^ { \prime } = 7 x ^ { 3 } y$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - y ^ { \prime \prime } = e ^ { 2 x }$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + 25 y = x$

Graph the particular solution and several other solutions. $2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x$

Solve the differential equation. $4 y ^ { \prime \prime } + y = 0$

A spring with a mass of 2 kg has damping constant 14, and a force of $4.8$ N is required to keep the spring stretched $0.4$ m beyond its natural length. Find the mass that would produce critical damping.

Use power series to solve the differential equation.. $\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } + y = \sec x , \frac { \pi } { 4 } < x < \frac { \pi } { 2 }$

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } - y ^ { \prime } - 2 y = 0 , y ( - 1 ) = 1 , y ( 1 ) = 0$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } - 3 y ^ { \prime } = \sin 9 x$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } - 4 y ^ { \prime } + 5 y = 7 e ^ { - x }$

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 5 y ^ { \prime } - 36 y = 0 , y ( 0 ) = 0 , y ( 2 ) = 1$

Solve the differential equation. $y ^ { \prime \prime } - 8 y ^ { \prime } + 25 y = 0$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 3 y ^ { \prime } = e ^ { 3 x }$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } + 5 y ^ { \prime } + 6 y = x ^ { 2 }$

A spring has a mass of $1$ kg and its damping constant is $c = 10$ . The spring starts from its equilibrium position with a velocity of $1$ m/s. Graph the position function for the spring constant $k = 20$ .