Question

Use the method of undetermined coefficients to find one solution of y''+4y'-3y=(7t^2-2t-8)e^(3t)
Note that the method finds a specific solution, not the general one.

can any one help me with this one i know how to solve the homoguenouse eq but i had problem with this one ?????? plzzz help

Answer 1

i found out \[yc= c1\times e ^{-2-\sqrt{7}} + c2 \times e ^{-2+\sqrt{7}}\]

Answer 2

but i dont know how to find yp1, yp2

Answer 3

First of all, there is only one particular solution (or at least only one up to addition of scalar multiples of the homogeneous solutions, y1 and y2.

Now, to find a particular solution you need to write down a guess for the form of that equation. You do that by looking at the form of the function on the RHS of the equation, the input/ the"driving" function/the "inhomogeneous function". Break it down into single terms and find all their derivatives.

In this case it's easy. You have, factoring out constants for now,
t^2e^3t hence the derivatives are all linear combinations of t^2e^3t, te^3t and e^3t;
te^3t and hence terms te^3t, e^3t; and
e^3t

Answer 4

Therefore our guess for the particular solution is

yp = At^2.e^3t + Bt.e^3t + Ce^3t

Answer 5

Substitute that into the full equation, the inhomogeneous equation and find three equations in A, B and C. Then solve those equations.

Answer 6

do u mean the eq of yp =ar^3+bt+c

Answer 7

but what about e^3t

Answer 8

No, I mean

yp = At^2.e^3t + Bt.e^3t + Ce^3t

Answer 9

and then i ll find yp' and yp'' from this eq ????????

Answer 10

You must have seen an example of this method in your class.

step 1: write down the guess of the yp with undetermined coefficients
step 2: substitute that guess into the inhomogeneous ODE
step 3: solve for the coefficients

So yes, the next step is step 2: substitute into your equation.

Before you do however, you should go back to your notes or text book and see an example worked.

Answer 11

ok thanks i ll try it know

Answer 12

coz my eq was wrong that why i couldnt solve it but let me try now

Answer 13

i got y= e^(-2+(7)^(1/2))+e^(-2-(7)^(1/2))+((7/32)t^(2)*e^(3t))-((67/288)*t*e^(3t))-((817/2592)*e^(3t)) but its wrong i dont know where i missed it can u help

Answer 14

The particular solution of

\[ y''+4y'-3y=(7t^2-2t-8)e^{3t} \]

is

\[ y_p = e^{3t} \left( \frac{7}{18}t^2 - \frac{44}{81}t - \frac{271}{1458} \right) \]

Answer 15

still incorrect

Answer 16

That is the particular solution of that equation. So unless the equation is wrong, I don't understand why it's wrong.

Answer 17

the equation is right but when im plaguing in the solution it say incorrect i dont know whyyyyyyyyyyy??????????????????

Answer 18

remember the general solution is the sum of the complimentary/homogeneous solution and the particular solution.

So the particular solution by itself isn't the general answer.

Answer 19

So perhaps that's the problem.