You provided a second-order linear differential equation:

$2y'' + 5y' - 3y = 0$

This is a homogeneous linear differential equation with constant coefficients. The general approach is to solve it by finding the characteristic equation.

Step 1: Write the characteristic equation

To solve the differential equation, we first assume a solution of the form:

$y = e^{rt}$

Substituting $y = e^{rt}$, $y' = re^{rt}$, and $y'' = r^2 e^{rt}$ into the differential equation:

$2r^2 e^{rt} + 5r e^{rt} - 3e^{rt} = 0$

Since $e^{rt}$ is never zero, we can divide through by $e^{rt}$, which gives the characteristic equation:

$2r^2 + 5r - 3 = 0$

Step 2: Solve the characteristic equation

Now, we solve the quadratic equation:

$2r^2 + 5r - 3 = 0$

Using the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 2$, $b = 5$, and $c = -3$, we get:

$r = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)}$ $r = \frac{-5 \pm \sqrt{25 + 24}}{4}$ $r = \frac{-5 \pm \sqrt{49}}{4}$ $r = \frac{-5 \pm 7}{4}$

Thus, we have two distinct real roots:

$r_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}$ $r_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$

Step 3: Write the general solution

Since the characteristic equation has two distinct real roots $r_1 = \frac{1}{2}$ and $r_2 = -3$, the general solution to the differential equation is:

$y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t}$ $y(t) = c_1 e^{\frac{1}{2} t} + c_2 e^{-3t}$

This is the general solution to the given differential equation.

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Would you like more details on any specific part of the solution or have any other questions?

Related questions:

1. How do we handle differential equations with repeated roots?
2. What is the method to solve non-homogeneous differential equations?
3. Can we solve higher-order linear differential equations in a similar manner?
4. How does the Laplace transform method apply to solving differential equations?
5. What are the initial conditions required to solve a differential equation uniquely?

Tip:

To determine specific values for the constants $c_1$ and $c_2$, you need initial or boundary conditions for the problem.