78.3.4 problem 1.d

Internal problem ID [21000]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 1.d
Date solved : Thursday, October 02, 2025 at 07:01:17 PM
CAS classification : [`y=_G(x,y')`]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 18

Order:=5; 
ode:=diff(y(x),x) = (x^2+y(x)^2)^(1/2); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 1+x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {1}{24} x^{4}+\operatorname {O}\left (x^{5}\right ) \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 27

ode=D[y[x],x]==Sqrt[x^2+y[x]^2]; 
ic={y[0]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,4}]
 

\[ y(x)\to -\frac {x^4}{24}+\frac {x^3}{3}+\frac {x^2}{2}+x+1 \]

✓ Sympy. Time used: 0.289 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(x**2 + y(x)**2) + Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ y{\left (x \right )} = 1 + x + \frac {x^{2}}{2} + \frac {x^{3}}{3} - \frac {x^{4}}{24} + \frac {x^{5}}{60} + O\left (x^{6}\right ) \]