77.1.4 problem 15 (page 27)

Internal problem ID [17894]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 15 (page 27)
Date solved : Tuesday, January 28, 2025 at 11:09:51 AM
CAS classification : [_separable]

\begin{align*} x \sqrt {1+y^{2}}+y \sqrt {x^{2}+1}\, y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 1.028 (sec). Leaf size: 37

dsolve([x*sqrt(1+y(x)^2)+y(x)*sqrt(1+x^2)*diff(y(x),x)=0,y(0) = 1],y(x), singsol=all)
 

\[ y = \sqrt {3-2 \sqrt {x^{2}+1}\, \sqrt {2}+x^{2}+2 \sqrt {2}-2 \sqrt {x^{2}+1}} \]

✓ Solution by Mathematica

Time used: 0.305 (sec). Leaf size: 95

DSolve[{x*Sqrt[1+y[x]^2]+y[x]*Sqrt[1+x^2]*D[y[x],x]==0,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \sqrt {x^2-2 \sqrt {2} \sqrt {x^2+1}-2 \sqrt {x^2+1}+2 \sqrt {2}+3} \\ y(x)\to \sqrt {x^2+2 \sqrt {2} \sqrt {x^2+1}-2 \sqrt {x^2+1}-2 \sqrt {2}+3} \\ \end{align*}