To solve this initial value problem (IVP), we need to solve the second-order differential equation and then use the initial conditions to determine the constants $k_1$ and $k_2$.

The differential equation is: $\frac{d^2 y}{dt^2} + 7 \frac{dy}{dt} + 10y = 0$ with the initial conditions: $y(0) = 3 \quad \text{and} \quad y'(0) = 2$

Step 1: Solve the Characteristic Equation

The characteristic equation for the differential equation is: $r^2 + 7r + 10 = 0$

Solving this quadratic equation: $r = \frac{-7 \pm \sqrt{49 - 40}}{2} = \frac{-7 \pm 3}{2}$ This gives the roots: $r_1 = -2 \quad \text{and} \quad r_2 = -5$

Step 2: General Solution of the Differential Equation

Since we have two distinct real roots, the general solution is: $y(t) = k_1 e^{-2t} + k_2 e^{-5t}$

Step 3: Apply Initial Conditions

1. Using $y(0) = 3$: $y(0) = k_1 e^0 + k_2 e^0 = k_1 + k_2 = 3$

2. Using $y'(0) = 2$: First, find $y'(t)$: $y'(t) = -2k_1 e^{-2t} - 5k_2 e^{-5t}$ Then apply $t = 0$: $y'(0) = -2k_1 - 5k_2 = 2$

Step 4: Solve the System of Equations

We now have the system:

1. $k_1 + k_2 = 3$
2. $-2k_1 - 5k_2 = 2$

Solving for $k_1$ and $k_2$:

• From equation (1): $k_1 = 3 - k_2$

• Substitute $k_1$ in equation (2): $-2(3 - k_2) - 5k_2 = 2$ $-6 + 2k_2 - 5k_2 = 2$ $-3k_2 = 8$ $k_2 = -\frac{8}{3}$

• Substitute $k_2 = -\frac{8}{3}$ back into equation (1): $k_1 + \left(-\frac{8}{3}\right) = 3$ $k_1 = 3 + \frac{8}{3} = \frac{17}{3}$

Solution

The constants are: $k_1 = \frac{17}{3} \quad \text{and} \quad k_2 = -\frac{8}{3}$

Would you like a detailed explanation of any step?

Here are 5 related questions to deepen understanding:

1. How does the method change if the characteristic equation has complex roots?
2. What would happen if the initial conditions were different, e.g., $y(0) = 0$ and $y'(0) = 1$?
3. Can we verify the solution by plugging it back into the original differential equation?
4. How would the solution form change if the equation had repeated roots?
5. Why are initial conditions necessary to find a particular solution?

Tip: Always check if the characteristic equation has real, complex, or repeated roots, as this affects the general solution structure.