Finding the General Solution of the Differential Equation
To solve the second-order differential equation given by:
y”” + 4y” = 0
Step 1: Rewrite the Equation
We can rewrite this equation by letting y” = z. This transforms the original equation into:
z” + 4z = 0
Step 2: Characteristic Equation
The next step is to find the characteristic equation associated with the equation z” + 4z = 0. We assume a solution of the form:
z = e^{rt}
Substituting into the characteristic equation we get:
r^2 + 4 = 0
Step 3: Solve for r
Now we solve for r:
r^2 = -4 ⟹ r = extit{±2i}
Since we have imaginary roots, the general solution for z is:
z(t) = C_1 ext{cos}(2t) + C_2 ext{sin}(2t)
Step 4: Return to y
Recall that z = y”. So the next step is to integrate z to find y.
Integrating z(t) gives us:
y'(t) = C_1 imes 2 ext{sin}(2t) – C_2 imes 2 ext{cos}(2t) + C_3
Integrating again, we get:
y(t) = -C_1 ext{cos}(2t) – C_2 ext{sin}(2t) + C_3t + C_4
Step 5: Final General Solution
Thus, the general solution of the original differential equation is:
y(t) = -C_1 ext{cos}(2t) – C_2 ext{sin}(2t) + C_3t + C_4
Where C_1, C_2, C_3, and C_4 are constants determined by initial conditions.