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### Course: Differential equations>Unit 2

Lesson 3: Method of undetermined coefficients

# Undetermined coefficients 3

Another example where the nonhomogeneous part is a polynomial. Created by Sal Khan.

## Want to join the conversation?

• It might sound a little silly, but what happens when when you don't have 0 or f(t) on the right side, but a constant c? Do you treat c like an undetermined coeeficient, which would be then y=A, y'=1, y''=0 => A+1=c => A=1-c, if the D.E. was y''+y'+y-c=0?
• If you have just a constant on the right side, you can just move it to the other side and include it in your homogeneous portion.
• Did you half-drop a negative in substituting back in the original equation for y prime? I believe it should be +6Ax-3B
• No, I think it is correct as is. The "=" in the y prime equation just kind of looks like subtraction in the video. So substituting 2Ax + B into the -3y' part gives -6Ax - 3B.
• What if the right hand side is just x or 4x? Is it Ax+B then for the particular?
• Yes. The particular solution follows the form of a polynomial with degree one. Ax + B
• m so confused how would i guess for particular sol.?
• For a particular solution, you guess with the same function given on the Right Hand Side with an unknown coefficient in front.

For example, as in this one, Sal uses the base function of x^2, which is really x^2 + 0x + 0 and puts an unknown coefficient in front of each variable. Therefore, Ax^2 + Bx + C is his guess.

In past examples, if the Right Hand Side was 4e^(3x), we would guess Ae^(3x) If the Right Hand Side was sin(5x), we would guess Asin(5x) + B cos(5x)...Hope this helps!
• How do we solve a nonhomogeneous equation of this such. y''+4y'+4y=2cos^2x
• What would I do if I have to solve for Y(0)= 1.6 and Y'(0)=4. Do I apply this to the particular solution, or to the general for the homogenous or both?
• You would apply this to both. ( in this video, Sal starts writing "y=" at . This is "the solution") You have the "y=" as your first equation, then take the derivative of that "y'=" for your second equation.

All that is left is to plug 'n' chug.

Set the first equation "y=" to 1.6, and set the second equation "y'=" to 4.
Plug in 0 for your independent variable in "y=" equation and also the "y'=" equation.
You know have 2 equations and two unknown variables, Solve for one, plug into the other equation, solve for the second, Viola!
• Does anyone know what skill level these problems are? Would these be basic first year uni questions?
(1 vote)
• yeah , m 101% sure these videos would help and they obviously have skill level.... m studying at UET Pakistan which is of course engineering uni ... and m watching these videos for basic things and i have to say more for logics ..... bcz our maths Sir does not bother to tell us logic behind question and we are sitting like puppets ,
• Is it compulsory to add the general solution of homogeneous eqn in the final answer? Because the examples in the previous videos, doesn't have one.
(1 vote)
• Yes, the most general solution is always the sum of the homogenous solution plus the particular solution. If you limit the answer to only the particular solution, then you are ignoring a whole family of functions that are also solutions, and when the times comes to evaluate initial conditions, you won't be able to satisfy them.