Differential equations - 2nd order homogenous eq'n w/ unknown

In summary, the equation t d^2y/dt^2 - (1+3t) dy/dt + 3y = 0 has a solution of the form e^ct, for some constant c. Using the reduction of order method, the general solution is y(t) = c1(1+3t) + c2e^(3t), where c1 and c2 are constants.
  • #1
braindead101
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Given that the equation
t d^2y/dt^2 - (1+3t) dy/dt + 3y = 0. has a solution of the form e^ct, for some constant c, find the general solution (The answer is y(t) = c1(1+3t) + c2e^(3t)

Edit: I finished this question as i figured it out. but when i come down to the last step, i get this
y1(t) = e^3t
y2(t) = -1/3t - 1/9

y(t) = c1y1(t) + c2y2(t)
y(t) = c1e^3t + c2(-1/3t-1/9)
y(t) = c1e^3t + -1/9c2(3t+1)

can i bet c3 = -1/9c2?
so it's
y(t) = c1e^3t + c3(3t+1)
 
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  • #2
braindead101 said:
I havn't done a question like this, so I don't know where to start. should i divide everything by t and do reduction of order?

Why divide by t?

Are you familiar with how the reduction of order method woks? This site should help
http://tutorial.math.lamar.edu/AllBrowsers/3401/ReductionofOrder.asp
 
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  • #3
sorry i figured out how to do it,
can u confirm whether i am allowed to do the last step? i don't see why not but just incase
 
  • #4
braindead101 said:
can u confirm whether i am allowed to do the last step? i don't see why not but just incase

Yes, it's right.
 

FAQ: Differential equations - 2nd order homogenous eq'n w/ unknown

1. What is a second order homogeneous differential equation?

A second order homogeneous differential equation is a mathematical equation that involves a function and its derivatives up to the second order in terms of the independent variable. The equation is considered homogeneous when all the terms are of the same degree and do not contain any constant terms.

2. What does it mean for a differential equation to be homogenous?

A differential equation is considered homogeneous when all the terms in the equation are of the same degree and do not contain any constant terms. This means that the equation can be written in terms of a single variable and its derivatives, without any additional constants or parameters.

3. How do you solve a second order homogeneous differential equation?

To solve a second order homogeneous differential equation, you can use techniques such as separation of variables, substitution, or the method of undetermined coefficients. You will need to identify the type of equation and apply the appropriate method to find the general solution.

4. What are the initial conditions for a second order homogeneous differential equation?

The initial conditions for a second order homogeneous differential equation are the values of the dependent variable and its first derivative at a specific point in the domain of the equation. These initial conditions are used to find the particular solution of the equation.

5. What are the applications of second order homogeneous differential equations?

Second order homogeneous differential equations have various applications in physics, engineering, and other scientific fields. They are commonly used to model systems that involve acceleration, such as the motion of a spring or a pendulum. They are also used in circuit analysis and in the study of oscillations and vibrations.

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