Internal problem ID [7212]
Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}=x -1} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 460
dsolve((1+x^2)*diff(y(x),x$2)+1+diff(y(x),x)^2=x,y(x), singsol=all)
\[ y \left (x \right ) = -\left (\int \frac {-\left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \left (x +i\right ) \left (\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} \sqrt {-1+i}\, \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right )-4 \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} \left (x +i\right ) \sqrt {-1+i}\, c_{1} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right )-8 \left (\operatorname {HeunCPrime}\left (0, -i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) c_{1} \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}-\frac {\operatorname {HeunCPrime}\left (0, i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) \left (\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}}{4}\right ) \left (i x +1\right )}{\left (4 \left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} c_{1} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right ) \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}-\left (\frac {1}{2}-\frac {i x}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {1}{2}+\frac {i x}{2}\right ) \left (\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}\right ) \left (x +i\right )}d x \right )+c_{2} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(1+x^2)*y''[x]+1+(y'[x])^2==x,y[x],x,IncludeSingularSolutions -> True]
Not solved