60.1.513 problem 516
Internal
problem
ID
[10527]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
516
Date
solved
:
Monday, January 27, 2025 at 08:51:53 PM
CAS
classification
:
[[_homogeneous, `class A`]]
\begin{align*} \left (x^{2}+y^{2}\right ) f \left (\frac {x}{\sqrt {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: 1.772 (sec). Leaf size: 141
dsolve((y(x)^2+x^2)*f(x/(y(x)^2+x^2)^(1/2))*(diff(y(x),x)^2+1)-(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\begin{align*}
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a} f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-\sqrt {-f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right ) \left (f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-1\right )}}{\left (\textit {\_a}^{2}+1\right ) f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}d \textit {\_a} +c_{1} \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a} f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )+\sqrt {-f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right ) \left (f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-1\right )}}{\left (\textit {\_a}^{2}+1\right ) f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}d \textit {\_a} +c_{1} \right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.114 (sec). Leaf size: 253
DSolve[-(-y[x] + x*D[y[x],x])^2 + f[x/Sqrt[x^2 + y[x]^2]]*(x^2 + y[x]^2)*(1 + D[y[x],x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right ) K[1]^2+f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} (K[1]-i) (K[1]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} K[1]+i \sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}\right )}dK[1]&=-\log (x)+c_1,y(x)\right ] \\
\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right ) K[2]^2+f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} (K[2]-i) (K[2]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} K[2]-i \sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}\right )}dK[2]&=-\log (x)+c_1,y(x)\right ] \\
\end{align*}