Question

Find the solution of the initial value problem: y′′−7y′=49t,y(0)=7,y′(0)=0.

Answer 1

The student is asked to solve an initial value problem involving a second-order linear differential equation with given initial conditions. This involves finding a complementary function for the homogeneous part, a particular solution for the nonhomogeneous part, and using initial conditions to solve for constants.

Step-by-step explanation:

The student is asked to solve the initial value problem with the given second-order linear differential equation y'' - 7y' = 49t, with initial conditions y(0) = 7 and y'(0) = 0. To solve for y(t), we proceed with the following steps:

1. First, find the complementary function (yc) by solving the homogeneous equation y'' - 7y' = 0.
2. Then, find the particular solution (yp) by using the method of undetermined coefficients for the nonhomogeneous part 49t.
3. Combine yc and yp to get the general solution.
4. Apply the initial conditions to determine the constants in the general solution.

Let's start by solving the homogeneous equation associated with the differential equation, which resembles a second-order linear equation with constant coefficients. The solution to the homogeneous equation would typically be in the form yc(t) = C1ert, where r are the roots of the characteristic equation.

To find the particular solution yp, we propose a solution of the form At + B, since the nonhomogeneous term is a first-degree polynomial. We can then differentiate yp and substitute into the original differential equation to solve for A and B.

Finally, by applying the initial conditions, we can solve for the constants C1 and C2 in the general solution y(t) = yc(t) + yp(t).

The full solution would thus provide y(t), the function of time t that satisfies the differential equation given the initial conditions.