60.7.195 problem 1786 (book 6.195)
Internal
problem
ID
[11784]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1786
(book
6.195)
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 }-2 \left ({y^{\prime }}^{2}+1\right ) \left (-y+x y^{\prime }\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.194 (sec). Leaf size: 91
dsolve((y(x)^2+x^2)*diff(diff(y(x),x),x)-2*(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 &= \frac {c_{1} -\sqrt {c_{1}^{2}+\left (4 i c_{2} x +2\right ) c_{1} -4 c_{2}^{2} x^{2}-4 i c_{2} x +1}+1}{2 c_{2}} \\
y &= \frac {c_{1} +1+\sqrt {c_{1}^{2}+\left (4 i c_{2} x +2\right ) c_{1} -4 c_{2}^{2} x^{2}-4 i c_{2} x +1}}{2 c_{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.379 (sec). Leaf size: 353
DSolve[-2*(-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]
\[
\text {Solve}\left [\int _1^{y(x)}\frac {1}{K[5]-x \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{x}}\frac {2}{K[4]^2+1}dK[4]\right ]}dK[5]-\int _1^x\left (\int _1^{y(x)}-\frac {\frac {2 K[5] \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{K[6]}}\frac {2}{K[4]^2+1}dK[4]\right ]{}^2+1\right )}{\left (\frac {K[5]^2}{K[6]^2}+1\right ) K[6]}-\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{K[6]}}\frac {2}{K[4]^2+1}dK[4]\right ]}{\left (K[5]-K[6] \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {K[5]}{K[6]}}\frac {2}{K[4]^2+1}dK[4]\right ]\right ){}^2}dK[5]+\frac {\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {y(x)}{K[6]}}\frac {2}{K[4]^2+1}dK[4]\right ]}{y(x)-K[6] \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[3]^2+1}dK[3]\&\right ]\left [c_1+\int _1^{\frac {y(x)}{K[6]}}\frac {2}{K[4]^2+1}dK[4]\right ]}\right )dK[6]=c_2,y(x)\right ]
\]