# How to Determine a Differential Equation from a Given Solution?

• Naeem
In summary, the conversation is about finding a homogeneous linear differential equation with constant coefficients that has a given solution of y=C1sin3x+C2cos3x. The idea is to differentiate both sides with respect to x to come up with an equation in dy/dx. The solution is d^2y/dx^2+9=0 since the original equation must have complex roots 3i and -3i. The characteristic equation is lambda^2+9=0 and by multiplying (lambda-3i)(lambda+3i) and equating it to 0, the two values for lambda are found. It is also mentioned that not every differential equation is a first order equation and that it is important to
Naeem
Q. Determine a homogeneous linear differential equation with constant coefficients having having the following solution:

y = C1sin3x + C2cos3x

My idea is to differntiate both sides with respect to x and come up with an equation in dy/dx

what else? can be done...

Is my idea correct.

Hello,

Since the solution is in the form y=ASin3x+BCos3x,
The original equation must have complex roots 3i and -3i.
Thus, a possible solution is d^2y/dx^2+9=0. =)

Not every differential equation is a first order equation!

estalniath said:
Hello,

Since the solution is in the form y=ASin3x+BCos3x,
The original equation must have complex roots 3i and -3i.
Thus, a possible solution is d^2y/dx^2+9=0. =)

Pay attention.The characteristic equation is

$$\lambda^{2}+9=0$$

,but the ODE is

$$\frac{d^{2}y}{dx^2}+9y=0$$

Okay?

Daniel.

Can somebody explain how they arrived at $$\lambda^{2}+9=0$$

I know that the two roots are 3i and -3i. I had figured out this already.

Suppose you were given $$\lambda^{2}+9=0$$

How would factor it , in order to find the two values for $$\lambda By multiplying $(\lambda-3i)(\lambda+3i)$ and equating it to 0...? Daniel. Yup I got it thanks! Thanks for pointing that out Daniel! I guess that I took the "y" there for granted every time I used the characteristic solution to get the [tex]y_h$$

By the way- this was clearly a simple problem because the given combination was clearly a solution to a linear equation with constant coefficients. It's not always that simple. In general, given a combination of functions with TWO "unknown constants", you form the simplest equation, involving differentials, the eliminates those constants.

If you did NOT recognize y= C1cos(3x)+ C2sin(3x) as coming from &lambda;= 3i and -3i, you could have done this:
Since you are seeking a differential equation: DIFFERENTIATE-
y'= -3 C1 sin(3x)+ 3 C2 cos(3x).
Since there are two unknown constants, DIFFERENTIATE AGAIN-
y"= -9 C1 cos(3x)- 9 C2 sin(3x).

Now do whatever algebraic manipulations you need to eliminate the two constants.

(In this example, of course, just add y" and 3y.)

## 1. What is a homogeneous solution?

A homogeneous solution is a solution to a differential equation where all the terms contain the dependent variable and its derivatives. In other words, it is a solution that satisfies the equation when all the variables are set to zero.

## 2. Why is finding a homogeneous solution important?

Finding a homogeneous solution is important because it allows us to solve more complex differential equations and understand the behavior of systems in the long run. It also helps us identify patterns and relationships between different variables.

## 3. How do you find a homogeneous solution?

To find a homogeneous solution, we first set all the variables in the differential equation to zero. Then, we solve the resulting equation to find the general solution. Finally, we add any initial conditions to find the specific solution.

## 4. Can a homogeneous solution be unique?

Yes, a homogeneous solution can be unique. In some cases, the initial conditions may determine a unique solution. However, in other cases, there may be multiple solutions that satisfy the same equation.

## 5. What are some real-life applications of finding a homogeneous solution?

Finding a homogeneous solution has various applications in physical sciences, engineering, and economics. For example, it can help predict the growth of populations, analyze the behavior of chemical reactions, and model the movement of objects under the influence of forces.

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