2.3.22 Problem 22

Internal problem ID [8880]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 22
Date solved : Sunday, March 30, 2025 at 01:45:33 PM
CAS classification : [[_2nd_order, _missing_y]]

✓ Maple. Time used: 0.034 (sec). Leaf size: 377

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

$$y=\int \frac{\sqrt{-1+i}{(\frac{ix}{2}+\frac{1}{2})}^{i\sqrt{-1+i}}(x+i){(\frac{1}{2}-\frac{ix}{2})}^{\frac{i\sqrt{-2+2\sqrt{2}}}{2}}\mathrm{hypergeom}([\frac{i\sqrt{-2+2\sqrt{2}}}{2},\frac{i\sqrt{-1+i}}{2}+\frac{\sqrt{1+i}}{2}+1],[i\sqrt{-1+i}+1],\frac{ix}{2}+\frac{1}{2})+4{(-\frac{1}{2}+\frac{ix}{2})}^{i\sqrt{-1+i}}\sqrt{-1+i}{(\frac{1}{2}-\frac{ix}{2})}^{\frac{\sqrt{2+2\sqrt{2}}}{2}}{c}_1(x+i)\mathrm{hypergeom}([\frac{\sqrt{2+2\sqrt{2}}}{2},\frac{\sqrt{2+2\sqrt{2}}}{2}+1],[-i\sqrt{-1+i}+1],\frac{ix}{2}+\frac{1}{2})+8(\mathrm{HeunCPrime}(0,-i\sqrt{-1+i},-1,0,\frac{1}{2}-\frac{i}{2},\frac{x-i}{x+i})c_1{(-\frac{1}{2}+\frac{ix}{2})}^{i\sqrt{-1+i}}-\frac{\mathrm{HeunCPrime}(0,i\sqrt{-1+i},-1,0,\frac{1}{2}-\frac{i}{2},\frac{x-i}{x+i}){(\frac{ix}{2}+\frac{1}{2})}^{i\sqrt{-1+i}}}{4})(ix+1)}{(x+i)(4{(\frac{1}{2}-\frac{ix}{2})}^{\frac{\sqrt{2+2\sqrt{2}}}{2}}{(-\frac{1}{2}+\frac{ix}{2})}^{i\sqrt{-1+i}}\mathrm{hypergeom}([\frac{\sqrt{2+2\sqrt{2}}}{2},\frac{\sqrt{2+2\sqrt{2}}}{2}+1],[-i\sqrt{-1+i}+1],\frac{ix}{2}+\frac{1}{2})c_1-{(\frac{1}{2}-\frac{ix}{2})}^{\frac{i\sqrt{-2+2\sqrt{2}}}{2}}{(\frac{ix}{2}+\frac{1}{2})}^{i\sqrt{-1+i}}\mathrm{hypergeom}([\frac{i\sqrt{-2+2\sqrt{2}}}{2},\frac{i\sqrt{-1+i}}{2}+\frac{\sqrt{1+i}}{2}+1],[i\sqrt{-1+i}+1],\frac{ix}{2}+\frac{1}{2}))}dx+c_2$$

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
 

✗ 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

✗ 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