76.1.11 problem 11

Internal problem ID [17239]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 11
Date solved : Monday, March 31, 2025 at 03:45:39 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} y^{\prime }&=4 \sqrt {x y} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 72

ode:=diff(y(x),x) = 4*(x*y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\left (48 c_1 \,x^{3}-27 y c_1 +3\right ) \sqrt {x y}-64 c_1 \,x^{5}+36 y c_1 \,x^{2}+4 x^{2}}{\left (16 x^{3}-9 y\right ) \left (4 x^{2}-3 \sqrt {x y}\right )} = 0 \]

✓ Mathematica. Time used: 0.155 (sec). Leaf size: 28

ode=D[y[x],x]==4*Sqrt[x*y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{36} \left (8 x^{3/2}+3 c_1\right ){}^2 \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.449 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = \frac {C_{1}^{2}}{4} - \frac {4 C_{1} \sqrt {x^{3}}}{3} + \frac {16 x^{3}}{9} \]