Question

Score on last try: 0 of 1 pts. See Details for more. You can retry this question below Solve:

y''-6y'+8y=16t+12
y(0)=0,y' (0)=−14
y(t)=26/8e²ᵗ-85/16⁴ᵗ+3/4t+33/16x

Answer 1

The student's task is to solve a second-order linear nonhomogeneous differential equation and verify if the provided solution meets the equation and initial conditions. One must differentiate the proposed function and substitute back into the equation to check correctness.

Step-by-step explanation:

The question asks to solve a second-order linear nonhomogeneous differential equation with initial conditions. The equation in question is y'' - 6y' + 8y = 16t + 12, and the initial conditions given are y(0) = 0 and y'(0) = -14. The provided solution is y(t) = 26/8e2t - 85/16e4t + 3/4t + 33/16, which includes terms that represent the solution to the homogeneous equation as well as the particular solution that accounts for the nonhomogeneous part (16t + 12).

To check the student's solution, we need to substitute the proposed function into the given differential equation and verify if it satisfies the equation and the given initial conditions. The verification process involves finding the first and second derivatives of y(t), substituting these derivatives and y(t) into the original equation, and finally evaluating the initial conditions.