60.1.512 problem 515

Internal problem ID [10526]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 515
Date solved : Monday, January 27, 2025 at 08:51:40 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.303 (sec). Leaf size: 134

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= \operatorname {RootOf}\left (x^{2}+\textit {\_Z}^{2}-f \left (\textit {\_Z}^{2}+x^{2}\right )\right ) \\ y &= \cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} -\int _{}^{\csc \left (\textit {\_Z} \right )^{2} x^{2}}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}}{\textit {\_a} \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}d \textit {\_a} +2 c_{1} \right )\right ) x \\ y &= \cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\int _{}^{\csc \left (\textit {\_Z} \right )^{2} x^{2}}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}}{\textit {\_a} \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}d \textit {\_a} +2 c_{1} \right )\right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.796 (sec). Leaf size: 1922

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

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