# Worksheet: Linear Homogeneous Differential Equations with Constant Coefficients

In this worksheet, we will practice solving second- and higher-order linear homogeneous differential equations with constant coefficients.

Q1:

Consider the differential equation . Suppose that a student determined the solution to be . Based upon this information, is the student correct?

• ANo, the student should have determined the solution to be .
• BNo, the student should have determined the solution to be .
• CNo, the student should have determined that there is no solution.
• DYes, the student did not make any errors.

Q2:

Find the general solution for the following higher-order differential equation: .

• A
• B
• C
• D

Q3:

Find the general solution for the following homogeneous ordinary differential equation with constant coefficients: .

• A
• B
• C
• D

Q4:

Apply the superposition principle to find the solution for .

• A
• B
• C
• D

Q5:

Find a particular solution for which passes through the origin and through the point .

• A
• B
• C
• D

Q6:

Suppose there are differentiable functions and , defined for all real numbers and satisfying and .

From the derivatives, what can you deduce about ?

• A is a constant function.
• B is an exponential function.
• C is a hyperbolic function.
• D always equals 0.
• E is a trigonometric function.

It is known that every function that satisfies is a combination for some constants and . Find all functions satisfying and , where .

• A and
• B and
• C and
• D and
• E and

Sketching graphs of the functions above suggest that for some constants and . Find these constants, where .

• A,
• B,
• C,
• D,
• E,

Q7:

Find the particular solution for the following homogeneous differential equation: where , , and .

• A
• B
• C
• D

Q8:

Find the solution for the following second-order ordinary differential equation: , where and .

• A
• B
• C
• D

Q9:

Solve the following differential equation under the conditions and :

• A
• B
• C
• D

Q10:

Solve the following differential equation under the conditions and :

• A
• B
• C
• D

Q11:

Solve the following differential equation under the conditions and :

• A
• B
• C
• D

Q12:

Solve the fourth-order differential equation under the conditions , , , and .

• A
• B
• C
• D

Q13:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

• A
• B
• C
• D
• E

Q14:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

• A
• B
• C
• D
• E

Q15:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

• A
• B
• C
• D
• E

Q16:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

• A
• B
• C
• D
• E

Q17:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

• A
• B
• C
• D
• E

Q18:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

• A
• B
• C
• D
• E