15.22.8 problem 8

Internal problem ID [3342]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 40, page 186
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 04:36:27 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\sqrt {1+x y} \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: 0.005 (sec). Leaf size: 16

Order:=4; 
ode:=diff(y(x),x) = (1+x*y(x))^(1/2); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y \left (x \right ) = 1+x +\frac {1}{4} x^{2}+\frac {1}{8} x^{3}+\operatorname {O}\left (x^{4}\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 20

ode=D[y[x],x]==Sqrt[1+x*y[x]]; 
ic={y[0]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,3}]
 

\[ y(x)\to \frac {x^3}{8}+\frac {x^2}{4}+x+1 \]

✓ Sympy. Time used: 0.526 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(x*y(x) + 1) + Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=4)
 

\[ y{\left (x \right )} = 1 + x + \frac {x^{2}}{4} + \frac {x^{3}}{8} + O\left (x^{4}\right ) \]