Amy has 1/2 hour to do 3 chores. She plans to spend the same fraction of an hour on each chore. She wants to use the number line
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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
Take
Supposing that
So we can write the ODE as
which is linear in
Integrate both sides with respect to
Substitute
Integrate the right hand side by parts using
You should end up with
and provided that we restrict