32.3.9 problem Exact Differential equations. Exercise 9.12, page 79

Internal problem ID [5807]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number : Exact Differential equations. Exercise 9.12, page 79
Date solved : Monday, January 27, 2025 at 01:19:48 PM
CAS classification : [[_homogeneous, `class A`], _exact, _dAlembert]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 19

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

\[ c_{1} +\left (x^{2}+y \left (x \right )^{2}\right )^{{3}/{2}}+y \left (x \right )^{3} = 0 \]

✓ Solution by Mathematica

Time used: 60.309 (sec). Leaf size: 2125

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

\begin{align*} y(x)\to -\frac {x^2 \sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}+x^2 \sqrt {\frac {2 e^{6 c_1}}{x^4}-12 x^2+\frac {3 \left (-5 x^6+4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}-\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}-\frac {2 e^{3 c_1} \left (-9+\frac {e^{6 c_1}}{x^6}\right )}{\sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}}}-e^{3 c_1}}{6 x^2} \\ y(x)\to \frac {x^2 \left (-\sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}\right )+x^2 \sqrt {\frac {2 e^{6 c_1}}{x^4}-12 x^2+\frac {3 \left (-5 x^6+4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}-\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}-\frac {2 e^{3 c_1} \left (-9+\frac {e^{6 c_1}}{x^6}\right )}{\sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}}}+e^{3 c_1}}{6 x^2} \\ y(x)\to \frac {x^2 \sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}-x^2 \sqrt {\frac {2 e^{6 c_1}}{x^4}-12 x^2+\frac {3 \left (-5 x^6+4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}-\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}+\frac {2 e^{3 c_1} \left (-9+\frac {e^{6 c_1}}{x^6}\right )}{\sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}}}+e^{3 c_1}}{6 x^2} \\ y(x)\to \frac {x^2 \sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}+x^2 \sqrt {\frac {2 e^{6 c_1}}{x^4}-12 x^2+\frac {3 \left (-5 x^6+4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}-\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}+\frac {2 e^{3 c_1} \left (-9+\frac {e^{6 c_1}}{x^6}\right )}{\sqrt {\frac {e^{6 c_1}}{x^4}-6 x^2+\frac {3 \left (5 x^6-4 e^{6 c_1}\right )}{\sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}+\frac {3 \sqrt [3]{-11 x^{12}+14 e^{6 c_1} x^6+2 \sqrt {\left (-x^6+e^{6 c_1}\right ) \left (x^6+e^{6 c_1}\right ){}^3}-2 e^{12 c_1}}}{x^2}}}}+e^{3 c_1}}{6 x^2} \\ \end{align*}