# How to solve the given differential equation

• Dao Tuat
In summary, the given equation 4y'' - 4y' - 3y=0, with initial conditions y(0)=1 and y'(0)=5, can be solved using the quadratic formula to find the roots of r=3/2 and r=-1/2. Substituting these values into the general solution, y = c1e^(3x/2) + c2e^(-x/2), and using the initial conditions to find c1 and c2, the final solution is y = [(±2sqrt(7))/3 + 5/3]e(3x/2) + [±2sqrt(7) - 5]e(-x/2).
Dao Tuat
Can someone give me some help, or at least tell me if I'm on the right track with this?

Heres what I was given:
4y'' - 4y' - 3y=0
y(0)=1
y'(0)=5

Heres what I did:
(sorry about the c1 and c2, I don't know how to make them show up as subsript)

4(r^2) - 4r - 3 = 0
r = 3/2 r=-1/2

y = c1e^(3x/2) + c2e^(-x/2)

1 = c1 + c2

y'' = [3c1e^(3x/2)]/2 - [c2e^(-x/2)]/2

5 = 3c1/2 - c2/2

c2 = ±2sqrt(7) - 5

c1 = 1/(±2sqrt(7) - 5) = [±2sqrt(7)]/3 + 5/3

y = [(±2sqrt(7))/3 + 5/3]e(3x/2) + [±2sqrt(7) - 5]e(-x/2)

Does this seem at all right?

Looks good to me!

## 1. How do I identify the type of differential equation?

To identify the type of differential equation, you need to look at the highest order derivative present in the equation. If the equation contains only first derivatives, it is a first-order differential equation. If it contains second derivatives, it is a second-order differential equation, and so on.

## 2. What is the general process for solving a differential equation?

The general process for solving a differential equation involves separating the variables, integrating both sides, and then solving for the constant of integration. This process may vary depending on the type of differential equation, but this is the basic approach.

## 3. What are the different methods for solving differential equations?

There are several methods for solving differential equations, including separation of variables, substitution, integrating factors, and power series. The method used depends on the type and complexity of the equation.

## 4. How do I know if my solution to a differential equation is correct?

You can check the validity of your solution by substituting it back into the original equation and verifying that it satisfies the equation. You may also compare your solution to other known solutions, if available.

## 5. Can differential equations be solved analytically or numerically?

Yes, differential equations can be solved using both analytical and numerical methods. Analytical solutions involve finding a mathematical expression for the solution, while numerical solutions use numerical methods to approximate the solution.

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