60.2.10 problem 586

Internal problem ID [10597]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 586
Date solved : Tuesday, January 28, 2025 at 04:55:29 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}} \end{align*}

✓ Solution by Maple

Time used: 0.080 (sec). Leaf size: 64

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

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

✓ Solution by Mathematica

Time used: 0.396 (sec). Leaf size: 975

DSolve[D[y[x],x] == (x*F[y[x]/Sqrt[1 + x^2]])/Sqrt[1 + x^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\left (-\frac {K[1] \sqrt {K[1]^2+1} F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^3}{y(x) \left (K[1]^2 F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2-y(x)^2\right )}-\frac {K[1] F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2}{K[1]^2 F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2-y(x)^2}+\frac {K[1] F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1} y(x)}\right )dK[1]+\int _1^{y(x)}\left (-\frac {\sqrt {x^2+1} F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )}{-x^2 F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2-F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2+K[2]^2}-\int _1^x\left (\frac {K[1] \sqrt {K[1]^2+1} \left (\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F''\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) K[1]^2}{\sqrt {K[1]^2+1}}-2 K[2]+\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F''\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^3}{K[2] \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )^2}+\frac {K[1] \sqrt {K[1]^2+1} F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^3}{K[2]^2 \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )}-\frac {3 K[1] F''\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2}{K[2] \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )}+\frac {K[1] \left (\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F''\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) K[1]^2}{\sqrt {K[1]^2+1}}-2 K[2]+\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F''\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2}{\left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )^2}-\frac {2 K[1] F''\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1} \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )}-\frac {K[1] F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1} K[2]^2}+\frac {K[1] F''\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\left (K[1]^2+1\right ) K[2]}\right )dK[1]-\frac {K[2]}{-x^2 F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2-F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2+K[2]^2}\right )dK[2]=c_1,y(x)\right ] \]