Question

Use the method of undetermined coefficients to solve the following differential equation: y ′′

+4y=4x Answer: y(x)= +C 1
+C 2
NOTE: The order of your answers is important in this problem. For example, webwork may expect the answer "A+B" but the answer you give is "B+A". Both answers are correct but webwork will only accept the former. Note: You can earn partial credit on this problem.

Answer 1

The solution to the differential equation y'' + 4y = 4x is:

y(x) = x.

To solve the given differential equation using the method of undetermined coefficients, we first assume a particular solution in the form of y_p(x) = Ax + B, where A and B are constants to be determined.

Now, let's find the first and second derivatives of y_p(x):

y_p'(x) = A

y_p''(x) = 0

Substituting these derivatives into the original differential equation, we have:

0 + 4(Ax + B) = 4x

Simplifying this equation, we get:

4Ax + 4B = 4x

Comparing the coefficients of x on both sides, we have:

4A = 4 (coefficient of x)

4B = 0 (constant term)

From the first equation, we find A = 1.

From the second equation, we find B = 0.

Therefore, the particular solution is:

y_p(x) = x

Since there are no complementary solutions (homogeneous solutions) in the given differential equation, the general solution is just the particular solution: y(x) = y_p(x) = x.

Thus, the solution to the differential equation y'' + 4y = 4x is:

y(x) = x.