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17calculus 17calculus
First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > exam B2

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Differential Equations - Exam B2

This page contains a complete differential equations exam with worked out solutions to all problems. This is exam 2 from semester B.

Exam Details

Tools

Time

1 hour

Calculators

no

Questions

11

Formula Sheet(s)

none

Total Points

100

Other Tools

none

 

Downloads

download one page list of these questions

download multiple page exam with space to work out the solutions

Instructions:
- This exam is in four main parts, labeled sections 1-4, with different instructions for each section.
- Show all your work.
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).
- Give exact answers.

Section 1

Find the general solution of each of the following differential equations. Each question in this section is worth 5 points.

Question 1

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

answer

solution

Question 2

\( y''-2y'+y=0 \)

answer

solution

Question 3

\( y''-2y'+5y=0 \)

answer

solution

Section 2

Find the general solution to each of the following differential equations using the method of undetermined coefficients. Each question in this section is worth 10 points.

Question 4

\(\displaystyle{ y''+4y=e^{2x} }\)

answer

solution

Question 5

\(\displaystyle{ y''-4y = e^{2x} }\)

answer

solution

Question 6

\(\displaystyle{ y''-2y'+y = (2x+1)e^x }\)

answer

solution

Section 3

Solve the following problems.

Question 7

[15 points] Use variation of parameters to find the general solution of \(\displaystyle{ y''-4y'+4y = \sec^2(x)e^{2x} }\)

answer

solution

Question 8

[10 points] Determine the second fundamental solution of \( x^2y''-6y=0 \), given \( y_1 = x^3 \)

answer

solution

Section 4

Solve the following initial value problems.

Question 9

[7 points] \( y''-4y=0; \)   \( y(0)=0, ~~~ y'(0)=1 \)

answer

solution

Question 10

[8 points] \( y''+4y = e^{2x}; \) \( y(0)=1, \) \( y'(0)=3 \)

answer

solution

Question 11

[15 points] \( y''+y'+1.25y = 5\sin(x); \) \( y(0)=y'(0)=0 \)

answer

solution

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