2 is the answer jeuuehehe
Answer:
it’s -93
Step-by-step explanation:
Answer:
The solution set is (-6, 5).
Step-by-step explanation:
-8х - 7y = 13
-3х – 3 y = 3
Multiply the first equation by 3 and the second by -7:
-24x - 21y = 39
21x + 21y = -21 Adding:
-3x = 18
x = -6.
Substitute fro x in equation 1:
-8(-6) - 7y = 13
-7y = 13 - 48 = -35
y = 5.
Use reduction of order. Given a solution
, look for a second solution of the form
.
Compute the first two derivatives of
:
![y_2 = x^2v \\\\ {y_2}' = x^2v' + 2xv \\\\ {y_2}'' = x^2v''+4xv' + 2v](https://tex.z-dn.net/?f=y_2%20%3D%20x%5E2v%20%5C%5C%5C%5C%20%7By_2%7D%27%20%3D%20x%5E2v%27%20%2B%202xv%20%5C%5C%5C%5C%20%7By_2%7D%27%27%20%3D%20x%5E2v%27%27%2B4xv%27%20%2B%202v)
Substitute them into the ODE:
![x^4 (x^2v'' + 4xv' + 2v) + x^3 (x^2v' + 2xv) - 4x^2 (x^2v) = 1 \\\\ x^6v'' + 5x^5v' = 1](https://tex.z-dn.net/?f=x%5E4%20%28x%5E2v%27%27%20%2B%204xv%27%20%2B%202v%29%20%2B%20x%5E3%20%28x%5E2v%27%20%2B%202xv%29%20-%204x%5E2%20%28x%5E2v%29%20%3D%201%20%5C%5C%5C%5C%20x%5E6v%27%27%20%2B%205x%5E5v%27%20%3D%201)
Now substitute
and you end up with a linear ODE:
![x^6w'+5x^5w=1](https://tex.z-dn.net/?f=x%5E6w%27%2B5x%5E5w%3D1)
Multiply through both sides by
(if you're familiar with the integrating factor method, this is it):
![x^5w'+5x^4w = \dfrac1x](https://tex.z-dn.net/?f=x%5E5w%27%2B5x%5E4w%20%3D%20%5Cdfrac1x)
Bear in mind that in order to do this, we require
. Just to avoid having to deal with absolute values later, let's further assume
.
Notice that the left side is the derivative of a product,
![\left(x^5w\right)' = \dfrac1x](https://tex.z-dn.net/?f=%5Cleft%28x%5E5w%5Cright%29%27%20%3D%20%5Cdfrac1x)
Integrate both sides with respect to
:
![x^5w = \displaystyle \int\frac{\mathrm dx}x \\\\ x^5w = \ln(x) + C_1](https://tex.z-dn.net/?f=x%5E5w%20%3D%20%5Cdisplaystyle%20%5Cint%5Cfrac%7B%5Cmathrm%20dx%7Dx%20%5C%5C%5C%5C%20x%5E5w%20%3D%20%5Cln%28x%29%20%2B%20C_1)
Solve for
:
![w = \dfrac{\ln(x)+C_1}{x^5}](https://tex.z-dn.net/?f=w%20%3D%20%5Cdfrac%7B%5Cln%28x%29%2BC_1%7D%7Bx%5E5%7D)
Solve for
by integrating both sides:
![v = \displaystyle \int \frac{\ln(x)+C_1}{x^5} \,\mathrm dx](https://tex.z-dn.net/?f=v%20%3D%20%5Cdisplaystyle%20%5Cint%20%5Cfrac%7B%5Cln%28x%29%2BC_1%7D%7Bx%5E5%7D%20%5C%2C%5Cmathrm%20dx)
Integrate by parts:
![\displaystyle f = \ln(x) + C_1 \implies \mathrm df = \frac{\mathrm dx}x \\\\ \mathrm dg = \frac{\mathrm dx}{x^5} \implies g = -\frac1{4x^4} \\\\ \implies v = -\frac{\ln(x)+C_1}{4x^4} + \frac14 \int \frac{\mathrm dx}{x^5} \\\\ v = -\frac{\ln(x)+C_1}{4x^4} - \frac1{16x^4} + C_2 \\\\ v = -\frac{4\ln(x)+C_1}{16x^4}+C_2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%20%3D%20%5Cln%28x%29%20%2B%20C_1%20%5Cimplies%20%5Cmathrm%20df%20%3D%20%5Cfrac%7B%5Cmathrm%20dx%7Dx%20%5C%5C%5C%5C%20%5Cmathrm%20dg%20%3D%20%5Cfrac%7B%5Cmathrm%20dx%7D%7Bx%5E5%7D%20%5Cimplies%20g%20%3D%20-%5Cfrac1%7B4x%5E4%7D%20%5C%5C%5C%5C%20%5Cimplies%20v%20%3D%20-%5Cfrac%7B%5Cln%28x%29%2BC_1%7D%7B4x%5E4%7D%20%2B%20%5Cfrac14%20%5Cint%20%5Cfrac%7B%5Cmathrm%20dx%7D%7Bx%5E5%7D%20%5C%5C%5C%5C%20v%20%3D%20-%5Cfrac%7B%5Cln%28x%29%2BC_1%7D%7B4x%5E4%7D%20-%20%5Cfrac1%7B16x%5E4%7D%20%2B%20C_2%20%5C%5C%5C%5C%20v%20%3D%20-%5Cfrac%7B4%5Cln%28x%29%2BC_1%7D%7B16x%5E4%7D%2BC_2)
Solve for
:
![\displaystyle \frac{y_2}{x^2} = -\frac{4\ln(x)+C_1}{16x^4}+C_2 \\\\ y_2 = -\frac{4\ln(x)+C_1}{16x^2} + C_2x^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By_2%7D%7Bx%5E2%7D%20%3D%20-%5Cfrac%7B4%5Cln%28x%29%2BC_1%7D%7B16x%5E4%7D%2BC_2%20%5C%5C%5C%5C%20y_2%20%3D%20-%5Cfrac%7B4%5Cln%28x%29%2BC_1%7D%7B16x%5E2%7D%20%2B%20C_2x%5E2)
But since
is already accounted for, the second solution is just
![\displaystyle y_2 = -\frac{4\ln(x)+C_1}{16x^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y_2%20%3D%20-%5Cfrac%7B4%5Cln%28x%29%2BC_1%7D%7B16x%5E2%7D)
Still, the general solution would be
![\displaystyle \boxed{y(x) = -\frac{4\ln(x)+C_1}{16x^2} + C_2x^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cboxed%7By%28x%29%20%3D%20-%5Cfrac%7B4%5Cln%28x%29%2BC_1%7D%7B16x%5E2%7D%20%2B%20C_2x%5E2%7D)
Answer:
-8
Step-by-step explanation:
The opposite of 8 is -8.