2.3.22 Problem 22

Internal problem ID [10154]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 22
Date solved : Monday, December 08, 2025 at 07:43:25 PM
CAS classification : [[_2nd_order, _missing_y]]

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

Entering second order ode missing \(y\) solverThis is second order ode with missing dependent variable \(y\). Let

\begin{align*} u(x) &= y^{\prime } \end{align*}

Then

\begin{align*} u'(x) &= y^{\prime \prime } \end{align*}

Hence the ode becomes

\begin{align*} \left (x^{2}+1\right ) u^{\prime }\left (x \right )+1+u \left (x \right )^{2}-x = 0 \end{align*}

Which is now solved for \(u(x)\) as first order ode.

Unknown ode type.

Unable to solve. Terminating.

2.3.22.1 ✓ Maple. Time used: 0.013 (sec). Leaf size: 457

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

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

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
 

2.3.22.2 ✗ 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

2.3.22.3 ✗ 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