56.1.14 problem 14

Internal problem ID [8726]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 14
Date solved : Wednesday, March 05, 2025 at 06:14:02 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {\ln \left (y^{2}+1\right )}{\ln \left (x^{2}+1\right )} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

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

\[ \int \frac {1}{\ln \left (x^{2}+1\right )}d x -\int _{}^{y}\frac {1}{\ln \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} = 0 \]

✓ Mathematica. Time used: 0.688 (sec). Leaf size: 48

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\log \left (K[1]^2+1\right )}dK[1]\&\right ]\left [\int _1^x\frac {1}{\log \left (K[2]^2+1\right )}dK[2]+c_1\right ] \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.666 (sec). Leaf size: 22

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

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\log {\left (y^{2} + 1 \right )}}\, dy = C_{1} + \int \frac {1}{\log {\left (x^{2} + 1 \right )}}\, dx \]