75.4.4 problem 49

Internal problem ID [16635]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 49
Date solved : Monday, March 31, 2025 at 03:02:37 PM
CAS classification : [_separable]

\begin{align*} 1+y^{2}&=x y^{\prime } \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 9

ode:=1+y(x)^2 = x*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \tan \left (\ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.242 (sec). Leaf size: 42

ode=(1+y[x]^2)==x*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ][\log (x)+c_1] \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}

✓ Sympy. Time used: 0.271 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + y(x)**2 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \tan {\left (C_{1} + \log {\left (x \right )} \right )} \]