To solve the given differential equation

$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 10y = 0,$

we need to find the general solution. Let's go through the steps:

1. Form the Characteristic Equation

Replace $y$, $\frac{dy}{dx}$, and $\frac{d^2y}{dx^2}$ with $r^0, r^1, r^2$, respectively:

$r^2 + 3r - 10 = 0.$

2. Solve the Quadratic Equation

The characteristic equation is quadratic. Solve it using the quadratic formula:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$

where $a = 1$, $b = 3$, and $c = -10$. Substituting these values:

$r = \frac{-3 \pm \sqrt{3^2 - 4(1)(-10)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 40}}{2} = \frac{-3 \pm \sqrt{49}}{2}.$

$r = \frac{-3 \pm 7}{2}.$

This gives two distinct real roots:

$r_1 = 2, \quad r_2 = -5.$

3. Write the General Solution

For distinct real roots $r_1$ and $r_2$, the general solution is:

$y(x) = C_1e^{r_1x} + C_2e^{r_2x}.$

Substitute $r_1 = 2$ and $r_2 = -5$:

$y(x) = C_1e^{2x} + C_2e^{-5x}.$

4. Select the Correct Option

The correct solution matches option (d):

$y = C_1e^{2x} + C_2e^{-5x}.$

Let me know if you'd like further explanation or additional examples!

Related Questions:

1. How do you solve characteristic equations with complex roots?
2. What happens if the roots of the characteristic equation are repeated?
3. Can we solve this equation using numerical methods instead?
4. How do boundary or initial conditions affect the solution?
5. How does the presence of a non-homogeneous term change the solution process?

Tip:

Always verify your roots carefully, as small errors in solving the quadratic can lead to incorrect solutions.