# How do I solve this differential equation?

• Ben-CS
In summary, The conversation discusses solving a differential equation involving the third derivative of y, with a given condition for b. The general solution is given in terms of complex constants and can be rewritten in the real field with new constants, A and B.

#### Ben-CS

I have decided to post an exercise and see what happens. You can post your own problems, too, if you wish. Have fun with this one!

Solve the following differential equation:

y&prime;&prime;&prime;&nbsp;=&nbsp;y

y(t) = a*exp(b*t) where b^3 = 1.

The general solution is of the form

y(x)=c1*exp(q1*x)+c2*exp(q2*x)+c3*exp(q3*x)

where c1, c2 and c3 are (complex) constant that can be fixed by initial conditions and
q1=1
q2=(-1+i*sqrt(3))/2
q3=(-1-i*sqrt(3))/2

This can be re-written in the real field as

y(x)=exp(x)* c1 + A*exp(-x/2)*cos(sqrt(3)*x+B) )

where A and B are new constants depending on c2 and c3 only.

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

To identify the type of differential equation, you will need to look at the highest order of the derivative and the type of functions involved. The three main types are: ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs).

## 2. What are the steps to solve a differential equation?

The general steps to solve a differential equation are: 1. Identify the type of differential equation. 2. Separate the variables. 3. Integrate both sides. 4. Solve for the constant of integration. 5. Check your solution by plugging it back into the original equation.

## 3. How do I handle initial conditions in a differential equation?

Initial conditions are often given in a differential equation to help determine the particular solution. You can use these conditions to solve for the constant of integration. Plug in the given values to the general solution and solve for the constant. This will give you the particular solution that satisfies the initial conditions.

## 4. Is there a specific method for solving each type of differential equation?

Yes, there are various methods for solving different types of differential equations. For example, Euler's method is commonly used for solving first-order ODEs, while separation of variables is used for solving first-order linear ODEs. For PDEs, methods like the method of characteristics or separation of variables can be used.

## 5. Are there any common mistakes to avoid when solving a differential equation?

Some common mistakes to avoid when solving a differential equation include: 1. Forgetting to check the solution by plugging it back into the original equation. 2. Not simplifying the solution enough. 3. Not accounting for the constant of integration. 4. Mixing up the independent and dependent variables. 5. Making calculation errors when integrating or solving for the constant.