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

Differential Equations Alpha List

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

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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 Downloads Time 1 hour Calculators no Questions 11 Formula Sheet(s) none Total Points 100 Other Tools none download one page list of these questions

Instructions:
- This exam is in four main parts, labeled sections 1-4, with different instructions for each section.
- 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).

Section 1

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

Question 1

$$y''-2y'-3y = 0$$

solution

Question 2

$$y''-2y'+y=0$$

solution

Question 3

$$y''-2y'+5y=0$$

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

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

solution

Question 5

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

solution

Question 6

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

solution

Section 3

Solve the following problems.

Question 7

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

solution

Question 8

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

solution

Section 4

Solve the following initial value problems.

Question 9

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

solution

Question 10

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

solution

Question 11

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