77.1.29 problem 46 (page 56)

Internal problem ID [17919]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 46 (page 56)
Date solved : Tuesday, January 28, 2025 at 08:28:11 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} \left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }&=\left (1+y^{2}\right )^{{3}/{2}} \end{align*}

✓ Solution by Maple

Time used: 0.677 (sec). Leaf size: 88

dsolve((y(x)-x)*sqrt(1+x^2)*diff(y(x),x)=(1+y(x)^2)^(3/2),y(x), singsol=all)
 

\[ \frac {\arctan \left (y\right ) y x -y c_{1} x -\sqrt {\frac {\left (x^{2}+1\right ) \left (1+y^{2}\right )}{\left (x y+1\right )^{2}}}\, y x +\arctan \left (y\right )-y-c_{1} +x -\sqrt {\frac {\left (x^{2}+1\right ) \left (1+y^{2}\right )}{\left (x y+1\right )^{2}}}}{x y+1} = 0 \]

✓ Solution by Mathematica

Time used: 1.140 (sec). Leaf size: 57

DSolve[(y[x]-x)*Sqrt[1+x^2]*D[y[x],x]==(1+y[x]^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\arctan (y(x))+\frac {\sqrt {y(x)^2+1} \left (x \sqrt {y(x)^2+1}-\sqrt {x^2+1}\right )}{x y(x)+1}-y(x)=c_1,y(x)\right ] \]