60.1.557 problem 560

Internal problem ID [10571]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 560
Date solved : Monday, January 27, 2025 at 09:08:09 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} a y \sqrt {1+{y^{\prime }}^{2}}-2 x y y^{\prime }+y^{2}-x^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 1.648 (sec). Leaf size: 1086

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

\begin{align*} -\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a}^{3}-2 y^{2} \textit {\_a} +\sqrt {a^{2} \left (y^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y^{2}+\textit {\_a}^{4}\right )}}{\left (-\textit {\_a}^{2}+y^{2}\right ) \sqrt {a^{2} \left (y^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y^{2}+\textit {\_a}^{4}\right )}-2 \textit {\_a} y^{4}+\left (-4 \textit {\_a}^{3}+2 \textit {\_a} \,a^{2}\right ) y^{2}-2 \textit {\_a}^{5}}d \textit {\_a} +\int _{}^{y}\frac {4 \left (\left (\frac {\left (\textit {\_f}^{2}-x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}}{2}-x^{5}-2 x^{3} \textit {\_f}^{2}+\left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x \right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (\left (-4 \textit {\_a}^{2}-a^{2}\right ) \textit {\_f}^{4}+\left (8 \textit {\_a}^{4}-2 \textit {\_a}^{2} a^{2}+a^{4}\right ) \textit {\_f}^{2}-5 \textit {\_a}^{4} a^{2}+12 \textit {\_a}^{6}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+a^{2} \textit {\_a} \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right ) \left (-2 \textit {\_a}^{4}-2 \textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right )}{\sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\, \left (2 a^{2} \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_a}^{5}-4 \textit {\_a}^{3} \textit {\_f}^{2}-2 \textit {\_a} \,\textit {\_f}^{4}-\textit {\_a}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\right )^{2}}d \textit {\_a} \right )+\frac {a^{2}}{4}-x^{2}\right ) \textit {\_f}}{\left (\textit {\_f}^{2}-x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}-2 x^{5}-4 x^{3} \textit {\_f}^{2}+2 \left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x}d \textit {\_f} +c_{1} &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {2 y^{2} \textit {\_a} -2 \textit {\_a}^{3}+\sqrt {a^{2} \left (y^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y^{2}+\textit {\_a}^{4}\right )}}{2 \textit {\_a} y^{4}+4 \textit {\_a}^{3} y^{2}-2 a^{2} \textit {\_a} y^{2}+2 \textit {\_a}^{5}+y^{2} \sqrt {a^{2} \left (y^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y^{2}+\textit {\_a}^{4}\right )}-\textit {\_a}^{2} \sqrt {a^{2} \left (y^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y^{2}+\textit {\_a}^{4}\right )}}d \textit {\_a} +\int _{}^{y}\frac {4 \left (\left (\frac {\left (-\textit {\_f}^{2}+x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}}{2}-x^{5}-2 x^{3} \textit {\_f}^{2}+\left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x \right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (\left (4 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{4}+\left (-8 \textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}-a^{4}\right ) \textit {\_f}^{2}+5 \textit {\_a}^{4} a^{2}-12 \textit {\_a}^{6}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+a^{2} \textit {\_a} \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right ) \left (-2 \textit {\_a}^{4}-2 \textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right )}{\sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\, \left (2 a^{2} \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_a}^{5}-4 \textit {\_a}^{3} \textit {\_f}^{2}-2 \textit {\_a} \,\textit {\_f}^{4}+\textit {\_a}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}-\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\right )^{2}}d \textit {\_a} \right )+\frac {a^{2}}{4}-x^{2}\right ) \textit {\_f}}{\left (-\textit {\_f}^{2}+x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}-2 x^{5}-4 x^{3} \textit {\_f}^{2}+2 \left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x}d \textit {\_f} +c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 45.126 (sec). Leaf size: 314

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

\begin{align*} \text {Solve}\left [\int _1^{y(x)}\left (\frac {4 x K[1]}{\left (x^2+K[1]^2\right )^2}-2 \int \left (-\frac {2 K[1]}{\left (x^2+K[1]^2\right )^2}-\frac {4 \left (x^2-K[1]^2\right ) K[1]}{\left (x^2+K[1]^2\right )^3}\right ) \, dx\right )dK[1]+2 \left (\int \frac {x^2-y(x)^2}{\left (x^2+y(x)^2\right )^2} \, dx-\frac {\sqrt {a^2 y(x)^2 \left (-a^2 y(x)^2+x^4+2 x^2 y(x)^2+y(x)^4\right )}}{a^2 y(x) \left (x^2+y(x)^2\right )}\right )&=c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\left (\frac {4 x K[2]}{\left (x^2+K[2]^2\right )^2}-2 \int \left (-\frac {2 K[2]}{\left (x^2+K[2]^2\right )^2}-\frac {4 \left (x^2-K[2]^2\right ) K[2]}{\left (x^2+K[2]^2\right )^3}\right ) \, dx\right )dK[2]+2 \left (\frac {\sqrt {a^2 y(x)^2 \left (-a^2 y(x)^2+x^4+2 x^2 y(x)^2+y(x)^4\right )}}{a^2 y(x) \left (x^2+y(x)^2\right )}+\int \frac {x^2-y(x)^2}{\left (x^2+y(x)^2\right )^2} \, dx\right )&=c_1,y(x)\right ] \\ \end{align*}