2.2.15 problem 19

Internal problem ID [675]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.3. Slope fields and solution curves. Page 26
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 04:05:30 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 5

ode:=diff(y(x),x) = ln(1+y(x)^2); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 0 \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 6

ode=D[y[x],x] == Log[1+y[x]^2]; 
ic=y[0]==0; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.288 (sec). Leaf size: 22

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

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\log {\left (y^{2} + 1 \right )}}\, dy = x + \int \limits ^{0} \frac {1}{\log {\left (y^{2} + 1 \right )}}\, dy \]