60.7.194 problem 1785 (book 6.194)
Internal
problem
ID
[11783]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1785
(book
6.194)
Date
solved
:
Tuesday, January 28, 2025 at 06:11:24 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
\begin{align*} \left (x^{2}+y^{2}\right ) y^{\prime \prime }-\left ({y^{\prime }}^{2}+1\right ) \left (-y+x y^{\prime }\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.177 (sec). Leaf size: 53
dsolve((y(x)^2+x^2)*diff(diff(y(x),x),x)-(diff(y(x),x)^2+1)*(x*diff(y(x),x)-y(x))=0,y(x), singsol=all)
\begin{align*}
y &= -i x \\
y &= i x \\
y &= \tan \left (\operatorname {RootOf}\left (\cos \left (\textit {\_Z} \right )^{2} {\mathrm e}^{-\frac {2 \left (\textit {\_Z} c_{1} i+i \textit {\_Z} +c_{1} c_{2} -c_{2} \right )}{c_{1} -1}}-x^{2}\right )\right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.301 (sec). Leaf size: 340
DSolve[(y[x] - x*D[y[x],x])*(1 + D[y[x],x]^2) + (x^2 + y[x]^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\frac {1}{K[5]-x \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{x}}\frac {1}{K[4]^2+1}dK[4]\right ]}dK[5]-\int _1^x\left (\int _1^{y(x)}-\frac {\frac {K[5] \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{K[6]}}\frac {1}{K[4]^2+1}dK[4]\right ]{}^2+1\right )}{\left (\frac {K[5]^2}{K[6]^2}+1\right ) K[6]}-\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{K[6]}}\frac {1}{K[4]^2+1}dK[4]\right ]}{\left (K[5]-K[6] \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{K[6]}}\frac {1}{K[4]^2+1}dK[4]\right ]\right ){}^2}dK[5]+\frac {\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {y(x)}{K[6]}}\frac {1}{K[4]^2+1}dK[4]\right ]}{y(x)-K[6] \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {y(x)}{K[6]}}\frac {1}{K[4]^2+1}dK[4]\right ]}\right )dK[6]=c_2,y(x)\right ]
\]