# Thread: [SOLVED] A differential equation

1. ## [SOLVED] A differential equation

I must do something wrong. I have to solve $\displaystyle f'(t)=t^2f^2(t)$ with the initial condition $\displaystyle f(0)=\frac{1}{2}$.
So I wrote $\displaystyle \frac{df}{dt}=t^2f^2(t) \Leftrightarrow \frac{df}{f^2(t)}=t^2dt \Leftrightarrow \int \frac{df}{f^2(t)dt}=$ $\displaystyle \int t^2 dt \Leftrightarrow -\frac{1}{f(t)}=\frac{x^3}{3}+C \Leftrightarrow f(t)=-\frac{3}{t^3}+C$. I cleary see that $\displaystyle f$ in $\displaystyle 0$ don't exist, so how could it be equal to $\displaystyle \frac{1}{2}$, since whatever value of $\displaystyle C$ won't satisfy the impossible.

2. Originally Posted by arbolis
-\frac{1}{f(t)}=\frac{x^3}{3}+C \Leftrightarrow f(t)=-\frac{3}{t^3}+C[/tex].
If $\displaystyle \frac{1}{f(t)} = \frac{t^3}{3} + C$ then $\displaystyle f(t) = \frac{3}{t^3+3C} = \frac{3}{t^3+k}$.

3. Originally Posted by arbolis
I must do something wrong. I have to solve $\displaystyle f'(t)=t^2f^2(t)$ with the initial condition $\displaystyle f(0)=\frac{1}{2}$.
So I wrote $\displaystyle \frac{df}{dt}=t^2f^2(t) \Leftrightarrow \frac{df}{f^2(t)}=t^2dt \Leftrightarrow \int \frac{df}{f^2(t)dt}=$ $\displaystyle \int t^2 dt \Leftrightarrow -\frac{1}{f(t)}=\frac{x^3}{3}+C \Leftrightarrow f(t)=-\frac{3}{t^3}+C$. I cleary see that $\displaystyle f$ in $\displaystyle 0$ don't exist, so how could it be equal to $\displaystyle \frac{1}{2}$, since whatever value of $\displaystyle C$ won't satisfy the impossible.
from $\displaystyle -\frac{1}{f(t)}=\frac{x^3}{3}+C$:

we have $\displaystyle f(t) = -\frac{1}{\frac{t^3}{3}+C}$

this is different from $\displaystyle f(t)=-\frac{3}{t^3}+C$