# Math Help - solve non linear ODE.

1. ## solve non linear ODE.

hello,
I have an ODE and am unable to solve it can somebody help:

dy/dt = -5*t*y^2 + 5/t - 1/t^2 , initial condition is y(1) = 1

I've look at various ways to solve but can't variable separable benoulli, integrating factor, exact etc but just can't seem to do it anyone can help.
Thank you.

2. Originally Posted by bubuer
hello,
I have an ODE and am unable to solve it can somebody help:

dy/dt = -5*t*y^2 + 5/t - 1/t^2 , initial condition is y(t) = 1

I've look at various ways to solve but can't variable separable benoulli, integrating factor, exact etc but just can't seem to do it anyone can help.
Thank you.
I believe this expression is a Ricatti DE. See Riccati equation - Wikipedia, the free encyclopedia

I think they are a little tricky to solve and unfortunately I do not have experience with them. One of the math giants here might reply to you

3. It's not Ricatti right now, because there's no y term. Question: where is the initial condition? You have y(t) = 1. Is that supposed to be y(0) = 1?

[EDIT]: See running-gag's post for a correction.

4. sorry, initial condition is y(1) = 1.

5. Hi

I confirm this is a Riccati equation

If you can find a particular solution $y_1$ then letting $y = y_1 + u$, you can find a Bernoulli equation for u
Équation de Riccati - Wikipédia

Here $y_1(t)=\frac{1}{t}$ is a particular solution

At the end I can find $y(t) = \frac{1}{Ke^{10t}-\frac{t}{2}-\frac{1}{20}}+\frac{1}{t}$

K can be found using the initial condition

6. Oh, right. You can have the coefficient of the y term be zero. Forgot about that.

7. so, it's a special case of the ricatti eqations i'll have a closer look at these.