Solved: "Analytical solution for a separable scalar nonlinear ODEx˙(t)=−x(t)1+x(t)4withx(0)=x0∈R"

Pelizzolaf40

Answered question

2022-03-15

Analytical solution for a separable scalar nonlinear ODE

\dot{x} (t) = - x (t) \sqrt{1 + x {(t)}^{4}} with x (0) = x_{0} \in R

Answer & Explanation

Talaminiu2d

Beginner2022-03-16Added 4 answers

Following the observations in the comments, here is my proposed answer. First, for $x \neq 0$ we have
$- \int \frac{d x}{x \sqrt{1 + x^{4}}}$
𝕔 $= \frac{1}{2} arccoth (\sqrt{x^{4} + 1}) + c$
$= \int_{0}^{t} d s = t .$
Using the initial condition $x (0) = x_{0} \neq 0$ and solving for c gives
$c = g (x_{0}) = \frac{1}{2} {arccoth}^{2} (\sqrt{1 + x_{0^{4}}}) .$
Solving for x and ignoring the imaginary roots yields
$x = \pm {(\coth^{2} (2 c - 2 t) - 1)}^{\frac{1}{4}}$
$= \pm ({csch}^{2} (2 g (x_{0}) - 2 t))^{\frac{1}{4}}$
Which leads to the solution of the ODE for all $x_{0} \in R$ being
$x (t) = sign (x_{0}) ({csch}^{2} (2 g (x_{0}) - 2 t))^{\frac{1}{4}} .$

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