12.4.3 problem 3

Internal problem ID [1610]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Existence and Uniqueness of Solutions of Nonlinear Equations. Section 2.3 Page 60
Problem number : 3
Date solved : Saturday, March 29, 2025 at 11:07:32 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\tan \left (x y\right ) \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 43

ode:=diff(y(x),x) = tan(x*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -i \operatorname {RootOf}\left (-\operatorname {erf}\left (\frac {\left (-x +\textit {\_Z} \right ) \sqrt {2}}{2}\right ) \sqrt {\pi }-\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +\textit {\_Z} \right )}{2}\right )+\sqrt {2}\, c_1 \right ) \]

✓ Mathematica. Time used: 0.277 (sec). Leaf size: 69

ode=D[y[x],x]==Tan[x*y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{2} \sqrt {\frac {\pi }{2}} e^{\frac {x^2}{2}} \left (\text {erfi}\left (\frac {y(x)-i x}{\sqrt {2}}\right )+\text {erfi}\left (\frac {y(x)+i x}{\sqrt {2}}\right )\right )=c_1 e^{\frac {x^2}{2}},y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -tan(x*y(x)) + Derivative(y(x), x) cannot be solved by the lie group method