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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 $ar^2 + br + c = r^2 - 9 = 0$. 2) The roots of the auxiliary equation are $3, -3$ (enter answers as a comma separated list). 3) A fundamental set of solutions is $e^{(3x)}, e^{(-3x)}$ (enter answers as a comma separated list). Using these we obtain the the complementary solution $y_c = c_1y_1 + c_2y_2$ for arbitrary constants $c_1$ and $c_2$. Next we seek a particular solution $y_p$ 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 $y_p = $ We then find the general solution as a sum of the complementary solution $y_c = c_1y_1 + c_2y_2$ and a particular solution: $y = y_c + y_p$. Finally you are asked to use the general solution to solve an IVP. 5) Given the initial conditions $y(0) = 2$ and $y'(0) = -6$ 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 $ar^2 + br + c = r^2 - 9 = 0$.
2) The roots of the auxiliary equation are $3, -3$ (enter answers as a comma separated list).
3) A fundamental set of solutions is $e^{(3x)}, e^{(-3x)}$ (enter answers as a comma separated list). Using these we obtain the the complementary solution $y_c = c_1y_1 + c_2y_2$ for arbitrary constants $c_1$ and $c_2$.
Next we seek a particular solution $y_p$ 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 $y_p = $
We then find the general solution as a sum of the complementary solution $y_c = c_1y_1 + c_2y_2$ and a particular solution: $y = y_c + y_p$. Finally you are asked to use the general solution to solve an IVP.
5) Given the initial conditions $y(0) = 2$ and $y'(0) = -6$ 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 = r^2 - 9 = 0.
2) The roots of the auxiliary equation are 3, -3 (enter answers as a comma separated list).
3) A fundamental set of solutions is e^(3x), e^(-3x) (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) = 2 and y'(0) = -6 find the unique solution to the IVP
y =
Help Sheet: Undetermined Coefficients Notes

Added by Victor M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Linda Hand

12/11/2022

Step 1

Since the right-hand side of the equation is a cubic polynomial, we guess that Yp is also a cubic polynomial. We write Yp = Ax^3 + Bx^2 + Cx + D. We then substitute Yp and its derivatives into the differential equation and equate coefficients to find A, B, C, and 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 = r^2-9 = 0. 2) The roots of the auxiliary equation are 3,-3 (enter answers as a comma separated list). 3) A fundamental set of solutions is e^(3x),e^(-3x) (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) = 2 and y'(0) = -6 find the unique solution to the IVP y = Help Sheet: Undetermined Coefficients Notes

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00:01 Okay, we want to solve this problem using undetermined coefficients.

00:05 I'm just going to go ahead and work it, and then you pick out what answers go in the blocks, or in the boxes.

00:12 Okay, so first we get r squared minus 9 equals 0.

00:17 So r minus 3, r plus 3 equals 0.

00:23 So r is 3 and negative 3.

00:25 So our first y is c1e to the 3t x, sorry, plus c2e to the negative 3x.

00:41 Okay, now that's why i'm c.

00:45 Now we want to do y to p.

00:46 We want to do undetermined coefficients.

00:49 So we have a cubed here.

00:51 So we're going to have to guess a x cubed plus bx squared plus cx plus d.

01:01 So the first derivative will be 3a x squared plus 2bx plus c, and the second derivative will be 6ax plus 2b.

01:16 And now it says the second derivative minus 9 times the function.

01:22 So the second derivative minus 9 times the function square plus cx plus d equals to 81x cubed.

01:43 So we have 6ax plus 2b minus 9 a x cubed, minus 9bx squared, minus 9cx, minus 9 d equals 81x cubed.

02:00 So we get minus 9a equals 81, and then minus 9b equals 0, since there's no x squared.

02:18 Minus 9c plus 6a equals 0.

02:29 And then 2b minus 9d equals 0.

02:39 Okay, from the first one, we get a is negative 9.

02:43 From the second one, b is 0.

02:47 Negative 9, c minus 54 equals 0...

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