2.8.2 Problem (b)

Internal problem ID [20990]
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 : Saturday, November 29, 2025 at 01:27:52 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=x +\sqrt {y^{2}+1} \\ y \left (0\right ) &= 1 \\ \end{align*}

Existence and uniqueness analysis

\begin{align*} y^{\prime }&=x +\sqrt {y^{2}+1} \\ y \left (0\right ) &= 1 \\ \end{align*}

This is non linear first order ODE. In canonical form it is written as

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

The \(x\) domain of \(f(x,y)\) when \(y=1\) is

\[ \{-\infty <x <\infty \} \]

And the point \(x_0 = 0\) is inside this domain. The \(y\) domain of \(f(x,y)\) when \(x=0\) is

\[ \{-\infty <y <\infty \} \]

And the point \(y_0 = 1\) is inside this domain. Now we will look at the continuity of

\begin{align*} \frac {\partial f}{\partial y} &= \frac {\partial }{\partial y}\left (x +\sqrt {y^{2}+1}\right ) \\ &= \frac {y}{\sqrt {y^{2}+1}} \end{align*}

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(x=0\) is

\[ \{-\infty <y <\infty \} \]

And the point \(y_0 = 1\) is inside this domain. Therefore solution exists and is unique.

Unknown ode type.

✗ 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);
 

\[ \text {No solution found} \]

Maple trace

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 inverse_Riccati 
trying an equivalence to an Abel ODE 
differential order: 1; trying a linearization to 2nd order 
--- trying a change of variables {x -> y(x), y(x) -> x} 
differential order: 1; trying a linearization to 2nd order 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
   -> Computing symmetries using: way = 3 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 5 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type
 

✓ 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