20.4.30 problem Problem 46

Internal problem ID [3665]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 46
Date solved : Sunday, March 30, 2025 at 02:02:49 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+4 x y&=4 x^{3} \sqrt {y} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(y(x),x)+4*x*y(x) = 4*x^3*y(x)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ -x^{2}+1-{\mathrm e}^{-x^{2}} c_1 +\sqrt {y} = 0 \]

✓ Mathematica. Time used: 0.165 (sec). Leaf size: 29

ode=D[y[x],x]+4*x*y[x]==4*x^3*Sqrt[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-2 x^2} \left (e^{x^2} \left (x^2-1\right )+c_1\right ){}^2 \]

✓ Sympy. Time used: 0.474 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**3*sqrt(y(x)) + 4*x*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1}^{2} e^{- 2 x^{2}} + 2 C_{1} x^{2} e^{- x^{2}} - 2 C_{1} e^{- x^{2}} + x^{4} - 2 x^{2} + 1 \]