Dylan is rollerblading along an 8 mile trail in the forest park she stopped for water after 2.4 miles. After another 3.1 miles,
1 answer:
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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
Consider the function First, factor it:
The x-intercepts are at points
1. From the attached graph you can see that
- function is positive for
- function is negative for
2. Since
the function is neither even nor odd.
3. The domain is the range is
