Second Order Linear Nonhomogeneous Differential Equations

In summary, the problem requires finding the solutions to the homogeneous problem, a particular solution to the nonhomogeneous problem, and using the initial conditions to find the coefficients and form the general solution. This can be done by finding the solutions to the homogeneous problem and using the method of undetermined coefficients to find a particular solution. The initial conditions can then be used to find the coefficients and form the general solution.
  • #1
Mikesgto
18
0

Homework Statement


y''+7y'=392sin(7t)+686cos(7t) with y(0)=4 and y'(0)=9


Homework Equations


No real relevant equations


The Attempt at a Solution


I assumed since the g(t) has function of both sine and cosine the solution would be both the real and non real parts of the solution to

y''+7y'=1078e^(7it)

my solution came out to be
77/24tcos(7t)+539/24tsin(7t)+i[77/24tsin(7t)-539/24tcos(7t)]
but given the initial conditions, i don't know what happen to my C's.
Also, if anyone could explain this method in a way that is easy to understand, I would appreciate it. I tried following the class notes, but it was of little help.
Thank you.
 
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  • #2
Mikesgto said:

Homework Statement


y''+7y'=392sin(7t)+686cos(7t) with y(0)=4 and y'(0)=9


Homework Equations


No real relevant equations


The Attempt at a Solution


I assumed since the g(t) has function of both sine and cosine the solution would be both the real and non real parts of the solution to

y''+7y'=1078e^(7it)

my solution came out to be
77/24tcos(7t)+539/24tsin(7t)+i[77/24tsin(7t)-539/24tcos(7t)]
but given the initial conditions, i don't know what happen to my C's.
Also, if anyone could explain this method in a way that is easy to understand, I would appreciate it. I tried following the class notes, but it was of little help.
Thank you.

1. Find the solutions to the homogeneous problem y'' + 7y' = 0.
2. Find a particular solution to the nonhomogeneous problem y'' + 7y' = 392sin(7t)+686cos(7t). yp = A sin(7t) + B cos(7t) is a good place to start.
3. Form the general solution to the nonhomogeneous equation. Use the initial conditions to find the coefficients of the functions in step 1.
 

Related to Second Order Linear Nonhomogeneous Differential Equations

1. What is a second order linear nonhomogeneous differential equation?

A second order linear nonhomogeneous differential equation is a mathematical equation that involves a second derivative of an unknown function, as well as other terms that are linearly related to the function and its derivatives. The term "nonhomogeneous" indicates that the equation includes a function that is not equal to zero, making it more complex than a homogeneous equation.

2. How do you solve a second order linear nonhomogeneous differential equation?

To solve a second order linear nonhomogeneous differential equation, you can use the method of undetermined coefficients or the method of variation of parameters. In the method of undetermined coefficients, you assume a solution in the form of a polynomial, exponential, or trigonometric function and then plug it into the equation to solve for the coefficients. In the method of variation of parameters, you find a particular solution by multiplying a general solution of the corresponding homogeneous equation by a function and then solving for the function.

3. What is the role of initial conditions in solving a second order linear nonhomogeneous differential equation?

Initial conditions, also known as boundary conditions, are necessary to find a specific solution to a second order linear nonhomogeneous differential equation. These conditions are usually specified as values of the function and its derivatives at a specific point, and they help to narrow down the possible solutions and determine the values of the coefficients.

4. Can a second order linear nonhomogeneous differential equation have multiple solutions?

Yes, a second order linear nonhomogeneous differential equation can have multiple solutions. This is because the general solution to such an equation contains two arbitrary constants, which can result in an infinite number of specific solutions depending on the initial conditions. Additionally, some types of equations, such as those with repeated roots, can have multiple solutions that are linearly independent.

5. What are some real-life applications of second order linear nonhomogeneous differential equations?

Second order linear nonhomogeneous differential equations have many applications in physics, engineering, and other fields. They can be used to model the motion of a spring-mass system, the oscillations of an electrical circuit, the growth and decay of populations, and the heat transfer in a system. They are also commonly used in control theory to study the behavior of systems over time and make predictions about their future states.

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