Question

Use the method of undetermined coefficients to find one solution of

y′′+2y′+2y=(10t+7)e⁻ᵗcos(t)+(11t+25)e⁻ᵗsin(t).
(It doesn't matter which specific solution you find for this problem.)

Answer 1

To solve the differential equation using the method of undetermined coefficients, we propose a particular solution that matches the form of the nonhomogeneous function, compute its derivatives, and substitute back into the equation to determine the coefficients. The particular solution is then used to find the complete solution.

Step-by-step explanation:

To find one solution to the differential equation y''+2y'+2y=(10t+7)e^{-t}cos(t)+(11t+25)e^{-t}sin(t) using the method of undetermined coefficients, we first assume a particular solution of the same form as the nonhomogeneous part. Given the trigonometric and exponential components, we expect the solution to require terms like te^{-t}cos(t), te^{-t}sin(t), e^{-t}cos(t), and e^{-t}sin(t).

Let's propose a particular solution of the form:

y_p(t) = e^{-t}[(A_1t + B_1)cos(t) + (A_2t + B_2)sin(t)],

where A_1, A_2, B_1, and B_2 are constants to be determined. We'll need to compute the first and second derivatives of y_p(t) and then substitute them back into the original differential equation to solve for these constants.

Once that is done, the constants can be plugged back into the form of y_p(t) to obtain the particular solution. This solution combined with the general solution of the associated homogeneous equation should give us the complete solution to the original differential equation.