Answer:
$3.19 for 1 gallon, and $25.52 for 8 gallons.
Answer:
15/4
Step-by-step explanation:
It would be 12 bags
Explanation:
24+3c=5c
24=2c
c=12
Every 100 centimeters is 1 meter, so the answer would be 2 meters
Answer:
The general solution is
![y= (C_{1}+C_{1}x) e^0x+(C_{3}+C_{4}x) e^x +\frac{1}{2} (e^x(x^2-2x+2)-e^x(2(x-1)+e^x(2))](https://tex.z-dn.net/?f=y%3D%20%28C_%7B1%7D%2BC_%7B1%7Dx%29%20e%5E0x%2B%28C_%7B3%7D%2BC_%7B4%7Dx%29%20e%5Ex%20%2B%5Cfrac%7B1%7D%7B2%7D%20%28e%5Ex%28x%5E2-2x%2B2%29-e%5Ex%282%28x-1%29%2Be%5Ex%282%29%29)
+ ![\frac{x^2}{2}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B2%7D)
Step-by-step explanation:
Step :1:-
Given differential equation y(4) − 2y''' + y'' = e^x + 1
The differential operator form of the given differential equation
comparing f(D)y = e^ x+1
The auxiliary equation (A.E) f(m) = 0
![m^4 -2m^3+m^2 = 0](https://tex.z-dn.net/?f=m%5E4%20-2m%5E3%2Bm%5E2%20%3D%200)
![m^2(m^2 -2m+1) = 0](https://tex.z-dn.net/?f=m%5E2%28m%5E2%20-2m%2B1%29%20%3D%200)
![(m^2 -2m+1) this is the expansion of (a-b)^2](https://tex.z-dn.net/?f=%28m%5E2%20-2m%2B1%29%20this%20is%20the%20expansion%20of%20%28a-b%29%5E2)
![m^2 =0 and (m-1)^2 =0](https://tex.z-dn.net/?f=m%5E2%20%3D0%20and%20%28m-1%29%5E2%20%3D0)
The roots are m=0,0 and m =1,1
complementary function is ![y_{c} = (C_{1}+C_{1}x) e^0x+(C_{3}+C_{4}x) e^x](https://tex.z-dn.net/?f=y_%7Bc%7D%20%3D%20%28C_%7B1%7D%2BC_%7B1%7Dx%29%20e%5E0x%2B%28C_%7B3%7D%2BC_%7B4%7Dx%29%20e%5Ex)
<u>Step 2</u>:-
The particular equation is ![\frac{1}{f(D)} Q](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bf%28D%29%7D%20Q)
P.I = ![\frac{1}{D^2(D-1)^2} e^x+1](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%5E2%28D-1%29%5E2%7D%20e%5Ex%2B1)
P.I = ![\frac{1}{D^2(D-1)^2} e^x+\frac{1}{D^2(D-1)^2}e^{0x}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%5E2%28D-1%29%5E2%7D%20e%5Ex%2B%5Cfrac%7B1%7D%7BD%5E2%28D-1%29%5E2%7De%5E%7B0x%7D)
P.I = ![I_{1} +I_{2}](https://tex.z-dn.net/?f=I_%7B1%7D%20%2BI_%7B2%7D)
![\frac{1}{D^2} (\frac{x^2}{2!} )e^x + \frac{1}{D^{2} } e^{0x}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%5E2%7D%20%28%5Cfrac%7Bx%5E2%7D%7B2%21%7D%20%29e%5Ex%20%2B%20%5Cfrac%7B1%7D%7BD%5E%7B2%7D%20%7D%20e%5E%7B0x%7D)
![\frac{1}{D} means integration](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%7D%20means%20integration)
![\frac{1}{D^2} (\frac{x^2}{2!} )e^x = \frac{1}{2D} \int\limits {x^2e^x} \, dx](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%5E2%7D%20%28%5Cfrac%7Bx%5E2%7D%7B2%21%7D%20%29e%5Ex%20%3D%20%5Cfrac%7B1%7D%7B2D%7D%20%5Cint%5Climits%20%7Bx%5E2e%5Ex%7D%20%5C%2C%20dx)
applying in integration u v formula
![\int\limits {uv} \, dx = u\int\limits {v} \, dx - \int\limits ({u^{l}\int\limits{v} \, dx } )\, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7Buv%7D%20%5C%2C%20dx%20%3D%20u%5Cint%5Climits%20%7Bv%7D%20%5C%2C%20dx%20-%20%5Cint%5Climits%20%28%7Bu%5E%7Bl%7D%5Cint%5Climits%7Bv%7D%20%5C%2C%20dx%20%20%7D%20%29%5C%2C%20dx)
= ![\frac{1}{D^2(D-1)^2} e^x](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%5E2%28D-1%29%5E2%7D%20e%5Ex)
![\frac{1}{2D} (e^x(x^2)-e^x(2x)+e^x(2))](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2D%7D%20%28e%5Ex%28x%5E2%29-e%5Ex%282x%29%2Be%5Ex%282%29%29)
![\frac{1}{2} (e^x(x^2-2x+2)-e^x(2(x-1)+e^x(2))](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%28e%5Ex%28x%5E2-2x%2B2%29-e%5Ex%282%28x-1%29%2Be%5Ex%282%29%29)
![I_{2}](https://tex.z-dn.net/?f=I_%7B2%7D)
![= \frac{1}{D^2(D-1)^2}e^{0x}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B1%7D%7BD%5E2%28D-1%29%5E2%7De%5E%7B0x%7D)
![\frac{1}{D} \int\limits {1} \, dx= \frac{1}{D} x](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%7D%20%5Cint%5Climits%20%7B1%7D%20%5C%2C%20dx%3D%20%5Cfrac%7B1%7D%7BD%7D%20x)
again integration ![\frac{1}{D} x = \frac{x^2}{2!}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BD%7D%20x%20%3D%20%5Cfrac%7Bx%5E2%7D%7B2%21%7D)
The general solution is ![y = y_{C} +y_{P}](https://tex.z-dn.net/?f=y%20%3D%20y_%7BC%7D%20%2By_%7BP%7D)
![y= (C_{1}+C_{1}x) e^0x+(C_{3}+C_{4}x) e^x +\frac{1}{2} (e^x(x^2-2x+2)-e^x(2(x-1)+e^x(2))](https://tex.z-dn.net/?f=y%3D%20%28C_%7B1%7D%2BC_%7B1%7Dx%29%20e%5E0x%2B%28C_%7B3%7D%2BC_%7B4%7Dx%29%20e%5Ex%20%2B%5Cfrac%7B1%7D%7B2%7D%20%28e%5Ex%28x%5E2-2x%2B2%29-e%5Ex%282%28x-1%29%2Be%5Ex%282%29%29)
+ ![\frac{x^2}{2!}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E2%7D%7B2%21%7D)