2.3.22 problem 22

Internal problem ID [8556]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 22
Date solved : Wednesday, December 18, 2024 at 01:53:23 AM
CAS classification : [[_2nd_order, _missing_y]]

Solve

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

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = exp_sym 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(_b(_a)^2-_a+1)/(_a^2+1), _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   trying Bernoulli 
   trying separable 
   trying inverse linear 
   trying homogeneous types: 
   trying Chini 
   differential order: 1; looking for linear symmetries 
   trying exact 
   Looking for potential symmetries 
   trying Riccati 
   trying Riccati sub-methods: 
      <- Abel AIR successful: ODE belongs to the 2F1 2-parameter class 
<- differential order: 2; canonical coordinates successful 
<- differential order 2; missing variables successful`
 

Maple dsolve solution

Solving time : 0.021 (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)
 

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

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