79.8.2 problem (b)

Internal problem ID [21106]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 2. Theory of First order differential equations. Excercises XII at page 98
Problem number : (b)
Date solved : Thursday, October 02, 2025 at 07:08:24 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=x +\sqrt {1+y^{2}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✗ Maple

ode:=diff(y(x),x) = x+(1+y(x)^2)^(1/2); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 93

ode=D[y[x],x]==x+Sqrt[1+y[x]]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{6} \left (\frac {4 \sqrt {y(x)+1} \arctan \left (\frac {x}{\sqrt {-y(x)-1}}\right )}{\sqrt {-y(x)-1}}+2 \text {arctanh}\left (\frac {x}{2 \sqrt {y(x)+1}}\right )+2 \log \left (-x^2+y(x)+1\right )+\log \left (-x^2+4 y(x)+4\right )\right )=\frac {1}{6} (2 \log (2)+\log (8)),y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x - sqrt(y(x) + 1) + Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x - sqrt(y(x) + 1) + Derivative(y(x), x) cannot be solved by the lie group method