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.

---

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.

Explore the step-by-step solution for solving the second-order linear homogeneous differential equation 2y'' + 5y' - 3y = 0 using the characteristic equation method. Learn how to apply the quadratic formula to find distinct real roots and write the general solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation