Find the particular solution to the differential equation that satisfies when .

Question

Find the particular solution to the differential equation  that satisfies  when  . ​

Quizplus · Accepted Answer

The differential equation is of the form:

y' + 4y = 5x + 2

To find the particular solution, we need to use the method of undetermined coefficients.

First, we find the complementary solution by solving the homogeneous equation:

y' + 4y = 0

The characteristic equation is:

r + 4 = 0

So, the complementary solution is:

y_c(x) = c*e^(-4x)

where c is a constant.

Next, we assume the particular solution has the form:

y_p(x) = ax + b

where a and b are constants to be determined.

We substitute this into the differential equation to obtain:

y_p' + 4y_p = 5x + 2

a + 4ax + 4b = 5x + 2

Equating coefficients, we get:

4a = 5
4b = 2

So, a = 5/4 and b = 1/2.

Therefore, the particular solution is:

y_p(x) = (5/4)*x + 1/2

The general solution is:

y(x) = y_c(x) + y_p(x)

y(x) = c*e^(-4x) + (5/4)*x + 1/2

Now, we use the initial condition y(0) = 1 to find the value of c.

y(0) = c + 1/2 = 1

c = 1/2

Therefore, the particular solution that satisfies y(0)=1 and y'(0)=0 is:

y(x) = (1/2)*e^(-4x) + (5/4)*x + 1/2

Simplifying, we get:

y(x) = (5/4)*x + (1/2)*(e^(-4x) + 1)

Therefore, the best choice is C.

Find the Particular Solution to the Differential Equation That