30.6.37 problem 38

Internal problem ID [7572]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problems. page 79
Problem number : 38
Date solved : Tuesday, September 30, 2025 at 04:54:17 PM
CAS classification : [_separable]

\begin{align*} \sqrt {y}+\left (x^{2}+4\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 14

ode:=y(x)^(1/2)+(x^2+4)*diff(y(x),x) = 0; 
ic:=[y(0) = 4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (\arctan \left (\frac {x}{2}\right )-8\right )^{2}}{16} \]

✓ Mathematica. Time used: 0.123 (sec). Leaf size: 37

ode=Sqrt[y[x]] +( x^2+4 )*D[y[x],x]==0; 
ic={y[0]==4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{16} \left (\arctan \left (\frac {x}{2}\right )-8\right )^2\\ y(x)&\to \frac {1}{16} \left (\arctan \left (\frac {x}{2}\right )+8\right )^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 4)*Derivative(y(x), x) + sqrt(y(x)),0) 
ics = {y(0): 4} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants