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We consider the non-homogeneous problem y'' - 9y = 81x^3 First we consider the homogeneous problem y'' - 9y = 0 : 1) the auxiliary equation is ar^2 + br + c = = 0. 2) The roots of the auxiliary equation are (enter answers as a comma separated list). 3) A fundamental set of solutions is (enter answers as a comma separated list). Using these we obtain the the complementary solution yc = c1y1 + c2y2 for arbitrary constants c1 and c2. Next we seek a particular solution yp of the non-homogeneous problem y'' - 9y = 81x^3 using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find yp = We then find the general solution as a sum of the complementary solution yc = c1y1 + c2y2 and a particular solution: y = yc + yp. Finally you are asked to use the general solution to solve an IVP. 5) Given the initial conditions y(0) = -5 and y'(0) = -9 find the unique solution to the IVP y = Help Sheet: Undetermined Coefficients Notes

          We consider the non-homogeneous problem y'' - 9y = 81x^3
First we consider the homogeneous problem y'' - 9y = 0 :
1) the auxiliary equation is ar^2 + br + c =  = 0.
2) The roots of the auxiliary equation are (enter answers as a comma separated list).
3) A fundamental set of solutions is (enter answers as a comma separated list). Using these we obtain the the complementary solution yc = c1y1 + c2y2 for arbitrary constants c1 and c2.
Next we seek a particular solution yp of the non-homogeneous problem y'' - 9y = 81x^3 using the method of undetermined coefficients (See the link below for a help sheet)
4) Apply the method of undetermined coefficients to find yp =
We then find the general solution as a sum of the complementary solution yc = c1y1 + c2y2 and a particular solution: y = yc + yp. Finally you are asked to use the general solution to solve an IVP.
5) Given the initial conditions y(0) = -5 and y'(0) = -9 find the unique solution to the IVP
y =
Help Sheet: Undetermined Coefficients Notes

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We consider the non-homogeneous problem y” - 9y = 81x^3
First we consider the homogeneous problem y” - 9y = 0 :
1) the auxiliary equation is ar^2 + br + c = = 0.
2) The roots of the auxiliary equation are (enter answers as a comma separated list).
3) A fundamental set of solutions is (enter answers as a comma separated list). Using these we obtain the the complementary solution yc = c1y1 + c2y2 for arbitrary constants c1 and c2.
Next we seek a particular solution yp of the non-homogeneous problem y” - 9y = 81x^3 using the method of undetermined coefficients (See the link below for a help sheet)
4) Apply the method of undetermined coefficients to find yp =
We then find the general solution as a sum of the complementary solution yc = c1y1 + c2y2 and a particular solution: y = yc + yp. Finally you are asked to use the general solution to solve an IVP.
5) Given the initial conditions y(0) = -5 and y'(0) = -9 find the unique solution to the IVP
y =
Help Sheet: Undetermined Coefficients Notes

Added by Adam C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/29/2024

Step 1

1) The auxiliary equation for the homogeneous problem is given by: $m^2 + 9 = 0$ 2) The roots of the auxiliary equation are: $m = \pm 3i$ 3) A fundamental set of solutions is given by: $y_1 = e^{3it}$ and $y_2 = e^{-3it}$ The complementary solution is given Show more…

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We consider the non-homogeneous problem y'' - 9y = 81x^3 First we consider the homogeneous problem y'' - 9y = 0: 1) the auxiliary equation is ar^2 + br + c = 0. 2) The roots of the auxiliary equation are (enter answers as a comma separated list). 3) A fundamental set of solutions is (enter answers as a comma separated list). Using these we obtain the the complementary solution yc = c1y1 + c2y2 for arbitrary constants c1 and c2. Next we seek a particular solution yp of the non-homogeneous problem y'' - 9y = 81x^3 using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find yp = We then find the general solution as a sum of the complementary solution yc = c1y1 + c2y2 and a particular solution: y = yc + yp. Finally you are asked to use the general solution to solve an IVP. 5) Given the initial conditions y(0) = -5 and y'(0) = -9 find the unique solution to the IVP y = Help Sheet: Undetermined Coefficients Notes

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Transcript

00:01 A differential equation which is y double tacks minus of 9 y is equal to 81 x2.

00:08 So i need to find out the solution for this.

00:12 So this is given by d squared minus of 3.

00:15 And outside the bracket i have y.

00:17 So this part of the year refers to fd.

00:23 Oxetery equation can be given by therefore m equals to 0.

00:28 So this will be m square minus of 3 a 0.

00:31 It is minus of 9 which is equal to 0.

00:34 So then m squared is equal to 9 on m is equal to plus minus 3.

00:40 So hence it's c is given by 7 e power of 3x plus e to e power of minus of 3x.

00:49 That is given by the formula of 1 by fd into q.

00:54 Q part is the value which is on the right -hand side of the differential equation.

00:58 So thus pi will be equals to 1 by d square minus of 0.

01:02 9 and this is 81 x cube.

01:06 So for this i need to take out 9 outside the bracket.

01:10 So i can write this as 1 minus of d square by 9 and this will be power of minus of 1 and this is 81 .x2.

01:22 Okay.

01:23 If i could write down one step over here i could have written like this minus of 9.

01:28 This is 1 minus of b square by 9.

01:31 And this is 81 .6.

01:33 So what i've done is i've just taken that d squared by 9, 1 minus d squared by 9.

01:37 It was in the numerator...

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