69.1.15 problem 15

Internal problem ID [14168]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 06:18:33 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 9

dsolve(diff(y(x),x)=(1+y(x)^2)/(1+x^2),y(x), singsol=all)
 

\[ y = \tan \left (\arctan \left (x \right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.315 (sec). Leaf size: 55

DSolve[D[y[x],x]==(1+y[x]^2)/(1+x^2),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 ]\left [\int _1^x\frac {1}{K[2]^2+1}dK[2]+c_1\right ] \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}