The problem is to solve the second-order linear differential equation with constant coefficients:

$y'' - y' - 12y = 0,$ with the initial conditions: $y(0) = 0, \quad y'(0) = 14.$

Step 1: Solve the characteristic equation

The characteristic equation for this differential equation is obtained by replacing $y''$ with $r^2$, $y'$ with $r$, and $y$ with $1$: $r^2 - r - 12 = 0.$ Factoring this quadratic equation: $r^2 - r - 12 = (r - 4)(r + 3) = 0.$ Thus, the roots are: $r_1 = 4, \quad r_2 = -3.$

Step 2: General solution

The general solution for a second-order linear differential equation with distinct roots is: $y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}.$ Substituting the roots $r_1 = 4$ and $r_2 = -3$: $y(t) = C_1 e^{4t} + C_2 e^{-3t}.$

Step 3: Apply initial conditions

Condition 1: $y(0) = 0$

Substitute $t = 0$ into the solution: $y(0) = C_1 e^{4(0)} + C_2 e^{-3(0)} = C_1 + C_2.$ Since $y(0) = 0$, this gives: C_1 + C_2 = 0. \tag{1}

Condition 2: $y'(0) = 14$

Differentiate $y(t)$: $y'(t) = 4C_1 e^{4t} - 3C_2 e^{-3t}.$ Substitute $t = 0$: $y'(0) = 4C_1 e^{4(0)} - 3C_2 e^{-3(0)} = 4C_1 - 3C_2.$ Since $y'(0) = 14$, this gives: 4C_1 - 3C_2 = 14. \tag{2}

Step 4: Solve for $C_1$ and $C_2$

From equation (1): C_2 = -C_1. \tag{3} Substitute $C_2 = -C_1$ into equation (2): $4C_1 - 3(-C_1) = 14,$ $4C_1 + 3C_1 = 14,$ $7C_1 = 14 \implies C_1 = 2.$ From equation (3): $C_2 = -C_1 = -2.$

Step 5: Final solution

Substitute $C_1 = 2$ and $C_2 = -2$ into the general solution: $y(t) = 2e^{4t} - 2e^{-3t}.$

Final Answer:

$y(t) = 2e^{4t} - 2e^{-3t}.$

Would you like more details or have any questions?

Related Questions:

1. How do we solve second-order linear differential equations with repeated roots?
2. Can this solution be verified by substituting it back into the original equation?
3. How do initial conditions influence the solution of differential equations?
4. What happens if one of the roots of the characteristic equation is zero?
5. How would this problem change if the equation included a forcing term (e.g., $y'' - y' - 12y = f(t)$)?

Tip:

When solving differential equations, always verify your solution by substituting it back into the original equation to ensure correctness!