40.3.7 problem 23 (k)

Internal problem ID [6611]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 5. Equations of first order and first degree (Exact equations). Supplemetary problems. Page 33
Problem number : 23 (k)
Date solved : Monday, January 27, 2025 at 02:15:53 PM
CAS classification : [_exact]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 22

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

\[ \frac {\left (x^{2}+y^{2}\right )^{{3}/{2}}}{3}-x y+c_1 = 0 \]

✓ Solution by Mathematica

Time used: 27.320 (sec). Leaf size: 319

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

\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^4 x^2+\text {$\#$1}^2 \left (3 x^4-9 x^2\right )-18 \text {$\#$1} c_1 x+x^6-9 c_1{}^2\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^4 x^2+\text {$\#$1}^2 \left (3 x^4-9 x^2\right )-18 \text {$\#$1} c_1 x+x^6-9 c_1{}^2\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^4 x^2+\text {$\#$1}^2 \left (3 x^4-9 x^2\right )-18 \text {$\#$1} c_1 x+x^6-9 c_1{}^2\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^4 x^2+\text {$\#$1}^2 \left (3 x^4-9 x^2\right )-18 \text {$\#$1} c_1 x+x^6-9 c_1{}^2\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^4 x^2+\text {$\#$1}^2 \left (3 x^4-9 x^2\right )-18 \text {$\#$1} c_1 x+x^6-9 c_1{}^2\&,5\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+3 \text {$\#$1}^4 x^2+\text {$\#$1}^2 \left (3 x^4-9 x^2\right )-18 \text {$\#$1} c_1 x+x^6-9 c_1{}^2\&,6\right ] \\ \end{align*}