56.3.22 problem 22
Internal
problem
ID
[8880]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
22
Date
solved
:
Tuesday, January 28, 2025 at 03:37:04 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2}&=x \end{align*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 377
dsolve((1+x^2)*diff(y(x),x$2)+1+diff(y(x),x)^2=x,y(x), singsol=all)
\[
y = \int \frac {\left (x +i\right ) \sqrt {-1+i}\, \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}} \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )+4 \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} \left (x +i\right ) \sqrt {-1+i}\, \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} c_{1} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )+8 \left (\operatorname {HeunCPrime}\left (0, -i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {-i+x}{x +i}\right ) c_{1} \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}-\frac {\operatorname {HeunCPrime}\left (0, i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {-i+x}{x +i}\right ) \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}}}{4}\right ) \left (i x +1\right )}{\left (4 \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right ) \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} c_{1} -\left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}} \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )\right ) \left (x +i\right )}d x +c_{2}
\]
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[(1+x^2)*D[y[x],{x,2}]+1+(D[y[x],x])^2==x,y[x],x,IncludeSingularSolutions -> True]
Not solved