- #1
mathboy20
- 30
- 0
Homework Statement
Dear Friends,
Given the differential equation
[tex]\frac{1}{1+x^2} \frac{dx}{dt} = \frac{\alpha \cdot 2 \pi}{4} \cdot cos(t)[/tex]
with the condition that x(0) = 0
Then find the largest possible solution (this is how its stated) if either
[tex]\alpha_{1} = 1/2[/tex]
[tex]\alpha_{2} = 1[/tex]
The Attempt at a Solution
Don't I treat the above as a seperable differential equation?
Such that
[tex]{(\frac{1}{1+x^2})}^{-1})} \frac{dx}{dt} = {({\frac{\alpha \cdot 2 \pi}{4} \cdot cos(t)})}^{-1}[/tex]
[tex]1+x^2 dx = \frac{2}{\alpha \cdot \pi \cdot cos(t)} dt[/tex]
[tex]\int x^2 dx = \int \frac{2}{\alpha \cdot \pi \cdot cos(t)} -1 \ dt + C[/tex]
[tex] \frac{x^3}{3} = \frac{2 \cdot ln(\frac{-cos(t)}{sin(t)-1}- \alpha \cdot \pi \cdot (t-C)}{\alpha \cdot \pi}[/tex]
[tex] x^3 = 3 \cdot (\frac{2 \cdot ln(\frac{-cos(t)}{sin(t)-1}- \alpha \cdot \pi \cdot (t-C)}{\alpha \cdot \pi})[/tex]
[tex] x(t) = (3 \cdot (\frac{2 \cdot ln(\frac{-cos(t)}{sin(t)-1}- \alpha \cdot \pi \cdot (t-C)}{\alpha \cdot \pi})^{\frac{1}{3}}[/tex]
if I then insert x(0) = 0.
I get C = 0. Then this means that max solution is 0??
I just want to be sure. Can anybody please look at my result at then inform me. Have done this correctly?
Thank You in Advance
Fred