60.1.508 problem 511

Internal problem ID [10522]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 511
Date solved : Monday, January 27, 2025 at 08:45:37 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 5.201 (sec). Leaf size: 309

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

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

✓ Solution by Mathematica

Time used: 107.431 (sec). Leaf size: 3159

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

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