60.1.265 problem 266

Internal problem ID [10279]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 266
Date solved : Tuesday, January 28, 2025 at 04:26:40 PM
CAS classification : [`x=_G(y,y')`]

\begin{align*} \left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }-a \sqrt {\left (1+y^{2}\right )^{3}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.152 (sec). Leaf size: 193

dsolve((y(x)-x)*sqrt(x^2+1)*diff(y(x),x)-a*sqrt((y(x)^2+1)^3)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {-x +\sqrt {-a^{2} \left (x^{2}+1\right )^{2} \left (a^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ y &= \frac {-x -\sqrt {-a^{2} \left (x^{2}+1\right )^{2} \left (a^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ \frac {\left (\arctan \left (y\right )-c_{1} \right ) \sqrt {a^{2}-1}+\sqrt {\frac {\left (x^{2}+1\right ) \left (1+y^{2}\right ) a^{2}}{\left (x y+1\right )^{2}}}\, \cos \left (\arctan \left (x \right )-\arctan \left (y\right )\right ) \arctan \left (\frac {\cos \left (\arctan \left (x \right )-\arctan \left (y\right )\right )}{\sqrt {a^{2}-1}}\right )-a \arctan \left (\frac {\left (y-x \right ) \sqrt {a^{2}-1}}{\left (x y+1\right ) a}\right )}{\sqrt {a^{2}-1}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.814 (sec). Leaf size: 69

DSolve[(y[x]-x)*Sqrt[x^2+1]*D[y[x],x]-a*Sqrt[(y[x]^2+1)^3]==0,y[x],x,IncludeSingularSolutions -> True]
 

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