3.22 problem 22

Internal problem ID [7860]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 22
Date solved : Tuesday, October 22, 2024 at 02:46:52 PM
CAS classification : [[_2nd_order, _missing_y]]

Solve

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2}&=x \end{align*}

3.22.1 Maple step by step solution

3.22.2 Maple trace

Methods for second order ODEs:
 

3.22.3 Maple dsolve solution

Solving time : 0.014 (sec)
Leaf size : 377

dsolve((x^2+1)*diff(diff(y(x),x),x)+1+diff(y(x),x)^2 = x, 
       y(x),singsol=all)
 

\[ y = \int \frac {\left (x +i\right ) \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}} \sqrt {-1+i}\, \left (-\frac {i x}{2}+\frac {1}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )+4 \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} c_1 \left (-\frac {i x}{2}+\frac {1}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} \left (x +i\right ) \sqrt {-1+i}\, \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right )+8 \left (i x +1\right ) \left (\operatorname {HeunCPrime}\left (0, -i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) c_1 \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}}-\frac {\operatorname {HeunCPrime}\left (0, i \sqrt {-1+i}, -1, 0, \frac {1}{2}-\frac {i}{2}, \frac {x -i}{x +i}\right ) \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}}}{4}\right )}{\left (x +i\right ) \left (4 \left (-\frac {i x}{2}+\frac {1}{2}\right )^{\frac {\sqrt {2+2 \sqrt {2}}}{2}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {2+2 \sqrt {2}}}{2}, \frac {\sqrt {2+2 \sqrt {2}}}{2}+1\right ], \left [-i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right ) \left (-\frac {1}{2}+\frac {i x}{2}\right )^{i \sqrt {-1+i}} c_1 -\operatorname {hypergeom}\left (\left [\frac {i \sqrt {-2+2 \sqrt {2}}}{2}, \frac {i \sqrt {-1+i}}{2}+\frac {\sqrt {1+i}}{2}+1\right ], \left [i \sqrt {-1+i}+1\right ], \frac {i x}{2}+\frac {1}{2}\right ) \left (-\frac {i x}{2}+\frac {1}{2}\right )^{\frac {i \sqrt {-2+2 \sqrt {2}}}{2}} \left (\frac {i x}{2}+\frac {1}{2}\right )^{i \sqrt {-1+i}}\right )}d x +c_2 \]

3.22.4 Mathematica DSolve solution

Solving time : 0.0 (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