# What is the particular solution of $y'' - 4y' + 3y = 2t + e^t$

$y'' - 4y' + 3y = 2t + e^t$

Usually this will = 0. So I would just need to find the characteristic equation and factor it. In this case its $\ne 0$ so what do I do?

*What is the difference between Particular and General solution * ?

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The general solution is the solution with the right side 0, but covers all solutions of the homogeneous equation. The particular solution is any single solution of the whole equation, including the right side. All solutions to your equation are given by the sum of the particular solution you found and the general solution because the equation is linear. – Ross Millikan Feb 18 '13 at 0:42
A particular solution is one solution. The general solution is a form that usually involves arbitrary constants to cover all solutions. It is the sum of a particular solution, and the general solution of the homogeneous equation. – 1015 Feb 18 '13 at 0:42
@RossMillikan In this case, don't you call general solution the sum of a particular solution and the general solution of the homogeneous equation? That is, the general solution of the given equation. – 1015 Feb 18 '13 at 0:47
@40Plot Note there is the general solution of the homogeneous equation (rhs=0), and the general solution of the equation. They differ by a particular solution. – 1015 Feb 18 '13 at 0:55

Check that $$y_p(t)=2t+8-\frac{te^t}{2}$$ is a particular solution.

Now you say you can solve the homogeneous equation whose general solution is: $$y_h(t)=Ae^t+Be^{3t}.$$

So your general equation is: $$y(t)=y_p(t)+y_h(t)=2t+8-\frac{te^t}{2} +Ae^t+Be^{3t}.$$

Note: How did I come up with the particular solution? Looking at the rhs, we know that we can look for a particular solution of the form $y_p(t)=at+b+cte^t$.

Then we plug in and adapt the constants. Note that it's $cte^t$ and not $ce^t$ because $1-4+3=0$.

This is known as the method of undetermined coefficients.

See here for more details and examples: http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients

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The method of undetermined coefficients works. (The OP should look that up in his textbook.) Alternatively, you can use the method of variation of parameters. (After all, they even use that on other planets.) – GEdgar Feb 18 '13 at 1:31
@GEdgar Yes. And there are probably other planets where they have figured out the Riemann hypothesis. – 1015 Feb 18 '13 at 3:56
I refer to the movie "The Day the Earth Stood Still", where the alien (Michael Rennie) tells the einsteinesque earthling (Sam Jaffe) how to solve a math problem. – GEdgar Feb 18 '13 at 14:42