6.5.54 problem 53

Internal problem ID [1678]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 53
Date solved : Tuesday, September 30, 2025 at 04:58:59 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime }+\frac {3 y}{x}&=\frac {3 x^{4} y^{2}+10 x^{2} y+6}{x^{3} \left (2 x^{2} y+5\right )} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 18

ode:=diff(y(x),x)+3*y(x)/x = (3*x^4*y(x)^2+10*x^2*y(x)+6)/x^3/(2*x^2*y(x)+5); 
ic:=[y(1) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {-5+\sqrt {48 x +1}}{2 x^{2}} \]

✓ Mathematica. Time used: 0.62 (sec). Leaf size: 37

ode=D[y[x],x]+3/x*y[x]==(3*x^4*y[x]^2+10*x^2*y[x]+6)/(x^3*(2*x^2*y[x]+5)); 
ic=y[1]==1; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt {\frac {1}{x^2}} \sqrt {x^4 (48 x+1)}-5 x}{2 x^3} \end{align*}

✓ Sympy. Time used: 0.729 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + 3*y(x)/x - (3*x**4*y(x)**2 + 10*x**2*y(x) + 6)/(x**3*(2*x**2*y(x) + 5)),0) 
ics = {y(1): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {48 x + 1} - 5}{2 x^{2}} \]