# Natural and forced response of a differential equation

• MHB
• marcadams267
In summary, the conversation discusses the process for finding the natural and forced response of a differential equation. The natural response is found by setting the right side of the equation to 0 and solving for the general solution. The forced response is found using the method of "Undetermined Coefficients" and a solution of the form $x= Acos(3t)+ Bsin(3t)$. The general solution to the entire equation is then found by combining the natural and forced responses.
marcadams267
Greetings everyone, I am a bit new to differential equations and I am trying to solve for the natural and forced response of this equation:

dx/dt+4x=2sin(3t) ; x(0)=0

Now I know that for the natural response I set the right side of the equation equal to 0, so I get
dx/dt+4x=0, thus the characteristic equation is m+4=0 and I get -4 as the root and the solution to the natural response is Ae^-4t and since x(0)=0, A must be 0.
So the natural response is just 0.

However, I am not sure about how to get the forced response, which is when the right side of the equation is not set to 0. Any help is appreciated, thank you.

You have correctly found that the general solution to the "associated homogeneous equation" is $x(t)= Ae^{-4t}$. Now you need to find a single solution to the entire equation. The simplest method for getting a solution to the entire equation is "Undetermined Coefficients". Since the "non-homogeneous part" is $2sin(3x)$, "try" a solution of the form $x= Acos(3t)+ Bsin(3t)$. Then $dx/dt= -3Asin(3t)+ 3Bcos(3t)$ and the equation becomes $dx/dt+ 4x= -3Asin(3t)+ 3Bcos(3t)+ 4Acos(3t)+ 4Bsin(3t)= (3B+ 4A)cos(3t)+ (-3A+ 4B)sin(3t)= 2sin(3t)+ 0cos(3t)$ so we must have 3B+ 4A= 0 and -3A+ 4B= 2. Solve those two equations for A and B.

Multiply the first equation by 3: 9B+ 12A= 0. Multiply the second equation by 4: -12A+ 16B= 8. Adding those eliminates A: 7B= 2 so B= 2/7. Then 3B+ 4A= 6/7+ 4A= 0 so 4A= -6/7, A= -3/14. The general solution to the entire equation is $x(t)= Ae^{-4t}- (3/14) cos(3t)+ (2/7) sin(3t)$. NOW apply the "initial condition". x(0)=
A- 3/14= 0 so A= 3/14. The solution is $x(t)= (3/14)e^{-4t}- (3/14) cos(3t)+ (2/7) sin(3t)$.

## 1. What is the difference between natural and forced response in a differential equation?

The natural response of a differential equation is the solution that arises from the initial conditions and the inherent behavior of the system. It is independent of any external inputs or forces. The forced response, on the other hand, is the solution that is caused by external inputs or forces acting on the system.

## 2. How do I determine the natural response of a differential equation?

The natural response can be determined by solving the homogeneous version of the differential equation, which is obtained by setting all external inputs or forces to zero. This solution will depend on the initial conditions of the system.

## 3. Can the natural response of a differential equation be zero?

Yes, it is possible for the natural response to be zero if the initial conditions and the behavior of the system cancel each other out. This means that the system will only exhibit the forced response.

## 4. What factors affect the forced response of a differential equation?

The forced response is affected by the type and magnitude of the external inputs or forces acting on the system. It is also influenced by the initial conditions and the behavior of the system.

## 5. How can I determine the complete response of a differential equation?

The complete response is the sum of the natural and forced responses. It can be determined by adding the two solutions together, taking into account any overlap between the two. This will give a complete picture of the behavior of the system.

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