How can I solve my second order ODE?

Question

I am trying to solve

$\qquad y''(t)=\frac{1}{y(t)^3}-\Big(\frac{y(t)^4+1}{y(t)^5}\Big)y'(t)^2$

When I try to evaluate

Dsolve[{y''[t] == 1/y[t]^3 - ((y[t]^4 + 1)/y[t]^5)y'[t]^2}, y[t], t]

I just get back

Dsolve[{true}, y[t], t]

How can I fix this?

This "error" can be produced by inadvertently using the single = sign (Set command) instead of the double == sign (Equal command). To clear the condition either evaluate Remove[y] or quit the kernel and then re-evaluate the DSolve expression. — LouisB, Feb 22 '21 at 06:41
If what you got back had True instead of true, then related: (40314), (46214) — Michael E2, Feb 22 '21 at 14:03

score 1 · Answer 1 · answered Feb 22 '21 at 10:53

See this:

DSolve[{y''[t] == 1/y[t]^3 - ((y[t]^4 + 1)/y[t]^5) y'[t]^2}, y[t], t]
(*  {{y[t] -> 
   InverseFunction[
     Inactive[Integrate][-((Sqrt[2] E^(-(1/(4 K[1]^4))) K[1])/Sqrt[
        2 C[1] - ExpIntegralEi[-(1/(2 K[1]^4))]]), {K[1], 1, #1}] &][
    t + C[2]]}, {y[t] -> 
   InverseFunction[
     Inactive[Integrate][(Sqrt[2] E^(-(1/(4 K[2]^4))) K[2])/Sqrt[
       2 C[1] - ExpIntegralEi[-(1/(2 K[2]^4))]], {K[2], 1, #1}] &][
    t + C[2]]}}
*)

Have fun!

How can I solve my second order ODE?

1 Answers1