Rearrange the ODE as


Take

, so that

.
Supposing that

, we have

, from which it follows that


So we can write the ODE as

which is linear in

. Multiplying both sides by

, we have

![\dfrac{\mathrm d}{\mathrm dx}\bigg[e^{x^2}u\bigg]=x^3e^{x^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5Be%5E%7Bx%5E2%7Du%5Cbigg%5D%3Dx%5E3e%5E%7Bx%5E2%7D)
Integrate both sides with respect to

:
![\displaystyle\int\frac{\mathrm d}{\mathrm dx}\bigg[e^{x^2}u\bigg]\,\mathrm dx=\int x^3e^{x^2}\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5Be%5E%7Bx%5E2%7Du%5Cbigg%5D%5C%2C%5Cmathrm%20dx%3D%5Cint%20x%5E3e%5E%7Bx%5E2%7D%5C%2C%5Cmathrm%20dx)

Substitute

, so that

. Then

Integrate the right hand side by parts using



You should end up with



and provided that we restrict

, we can write
Positive infinity and negative infinity
Answer:
\simeq 14.94 billion dollars
Step-by-step explanation:
During the period 1994 - 2004, the 'National Income' ,(NI) of Australia grew about 5.2% per year (measured in 2003 U. S, dollars). In 1994 , the NI of Australia was $ 4 billion.
Now,
(2020 - 1994) = 26
Assuming this rate of growth continues, the NI of Australia in the year 2020 (in billion dollars) will be,
![4 \times[\frac{(100 + 5.2)}{100}}]^{26}](https://tex.z-dn.net/?f=4%20%5Ctimes%5B%5Cfrac%7B%28100%20%2B%205.2%29%7D%7B100%7D%7D%5D%5E%7B26%7D)
=![4 \times[\frac{105.2}{100}]^{26}](https://tex.z-dn.net/?f=4%20%5Ctimes%5B%5Cfrac%7B105.2%7D%7B100%7D%5D%5E%7B26%7D)
=\simeq 14.94 billion dollars (answer)
The best prediction for the number of customers is 5450.
To find this value, you just need to plug in 20 for x into the expression. Then, evaluate it using the order of operations.
8(20)^2 + 100(20) + 250
8(400) + 100(20) + 250
3200 + 2000 + 250
5450
The answer would be (-1,-6). Remember that the x-axis runs left to right and the y-axis runs up and down. Therefore, if you were to reflect it, just picture it being flipped over. If you go from (-1,+6) and you reflect it over the x-axis, both points are now in the bottom left quadrant where everything is negative. So, you get (-1,-6) The -1 doesn't change since it was already a -x value before and stayed in a negative area.