To solve the given initial value problem, we can use the method of solving second-order linear homogeneous differential equations with constant coefficients.
1. Find the characteristic equation:
The characteristic equation for the given differential equation is obtained by
substituting y = e^(rt) into the equation,
where r is the unknown constant:
9r^2 - 12r + 4 = 0
2. Solve the characteristic equation:
To solve the quadratic equation, we can factor it or use the quadratic formula. In this case, we can factor it as follows:
(3r - 2)(3r - 2) = 0
The repeated root is r = 2/3.
3. Write the general solution:
The general solution of the differential equation is:
y(t) = C1 * e^(2/3 * t) + C2 * t * e^2/3 * t)
4. Apply the initial conditions:
Using the given initial conditions, we can find the particular solution. Substitute t =
0, y(0) = 2, and y'(O) = -1 into the general
solution.
y(0) = C1* e^2/3 * 0) + C2 * 0 * e^(2/3 * 0)
= C1 = 2
y'(0) = (2/3) * C1 + C2 * 0 = 2/3 * 2 - C2 = -1
Solving the equations, we find C1 = 2 and
C2 = 8/3.
5. Write the final solution:
The final solution to the initial value problem is:
y(t) = 2 * e^(2/3 * t) + (8/3) * t * e^(2/3 * t)
To sketch the graph of the solution, you can plot points by substituting different values of t into the equation and connecting them with a smooth curve.