47.2.49 problem 45

Internal problem ID [7465]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 45
Date solved : Sunday, March 30, 2025 at 12:09:31 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Chini]

\begin{align*} 2 y^{\prime }+x&=4 \sqrt {y} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 98

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

\[ \frac {\left (-x^{2}+4 y\right ) \ln \left (\frac {x^{2}-4 y}{x^{2}}\right )+2 i \left (x^{2}-4 y\right ) \arctan \left (2 \sqrt {-\frac {y}{x^{2}}}\right )-4 i \sqrt {-\frac {y}{x^{2}}}\, x^{2}+4 \left (-c_1 +2 \ln \left (x \right )\right ) y+x^{2} \left (c_1 -2 \ln \left (x \right )-2\right )}{x^{2}-4 y} = 0 \]

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 49

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

\[ \text {Solve}\left [4 \left (\frac {4}{4 \sqrt {\frac {y(x)}{x^2}}+2}+2 \log \left (4 \sqrt {\frac {y(x)}{x^2}}+2\right )\right )=-8 \log (x)+c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE x/2 - 2*sqrt(y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method