50.3.22 problem 22
Internal
problem
ID
[10166]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
22
Date
solved
:
Tuesday, September 30, 2025 at 07:05:34 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2}&=x \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 377
ode:=(x^2+1)*diff(diff(y(x),x),x)+1+diff(y(x),x)^2 = x;
dsolve(ode,y(x), singsol=all);
\[
y = \int \frac {\sqrt {-1+i}\, \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}} \left (x +i\right ) \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )+4 \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} \sqrt {-1+i}\, \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} c_1 \left (x +i\right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )+8 \left (\operatorname {HeunCPrime}\left (0, -i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) c_1 \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}-\frac {\operatorname {HeunCPrime}\left (0, i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}}}{4}\right ) \left (i x +1\right )}{\left (x +i\right ) \left (4 \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right ) c_1 -\left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )\right )}d x +c_2
\]
✗ Mathematica
ode=(1+x^2)*D[y[x],{x,2}]+1+(D[y[x],x])^2==x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + (x**2 + 1)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2 + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : bad operand type for unary -: list