73.1.16 problem 3 (iv)

Internal problem ID [19789]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 3. Solutions of first-order equations. Exercises at page 47
Problem number : 3 (iv)
Date solved : Thursday, October 02, 2025 at 04:43:36 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (t^{2}-x^{2}\right ) x^{\prime }&=x t \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 19

ode:=(t^2-x(t)^2)*diff(x(t),t) = t*x(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \sqrt {-\frac {1}{\operatorname {LambertW}\left (-c_1 \,t^{2}\right )}}\, t \]

✓ Mathematica. Time used: 5.723 (sec). Leaf size: 56

ode=(t^2-x[t]^2)*D[x[t],t]==t*x[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {i t}{\sqrt {W\left (-e^{-2 c_1} t^2\right )}}\\ x(t)&\to \frac {i t}{\sqrt {W\left (-e^{-2 c_1} t^2\right )}}\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.780 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t*x(t) + (t**2 - x(t)**2)*Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = e^{C_{1} + \frac {W\left (- t^{2} e^{- 2 C_{1}}\right )}{2}} \]