Simplify differential equation

[salvomic]

(02-05-2015 07:21 AM)parisse Wrote:  [a]:=desolve(diff(y,t)=t^3*y^2+2t*y,t,y);
then lin(a) returns (-1/2*(t^2-1)+c_0*exp(-t^2))^-1
I don't think it's the same as your book.
simplify(diff(a,t)-(t^3*a^2+2t*a) returns 0, but
b:=1/(c*e^(t^2)-(2+t^2))
simplify(diff(b,t)-(t^3*b^2+2t*b))
does not return 0.
Perhaps you made a mistake in entering the equation, or the book solution is wrong.

I hadn't thought to lin()! ...

I've controlled in Wolfram Alpha, and I got y(t) = -(2 e^(t^2))/(c_1+e^(t^2) (t^2-1)):

\( y(t)=\frac{-2e^{t^{2}}}{c_1+e^{t^{2}}(t^{2}-1)} \)

(Wolfram web page says "Riccati equation", truly is a Bernoulli eq. form: y'=p(t)y^\alpha+q(t)y, as the Riccati one is a particular form with y'=p(t)y^2+q(t)y+r(t)...)
The same solution of Prime, if I'm not wrong, so could be an error in the book...

The book suggest to put \( z=y^{(1-\alpha)} \) and in this case give for \( z'=-(2tz+t^{3}) \) the solution \( z(t)=ce^{t^{2}}-(2+t^{2}) \), so the solution of the principal equation is \( \frac{1}{z(t)} \); however with Prime I get for it \( z(t)=\frac{1}{2}(-t^{2}+2*G_0e^{-t^{2}}+1) \)
...

Thank you for your patience!

Salvo

EDIT: rewrite something in LaTEX for clarity