76.12.7 problem Ex. 7

Internal problem ID [20057]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 7
Date solved : Thursday, October 02, 2025 at 05:17:21 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 3 y^{\prime }+\frac {2 y}{1+x}&=\frac {x^{3}}{y^{2}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 135

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

\begin{align*} y &= \frac {450^{{1}/{3}} {\left (\left (10 x^{6}+24 x^{5}+15 x^{4}+60 c_1 \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}}}{30 \left (x +1\right )^{2}} \\ y &= -\frac {450^{{1}/{3}} {\left (\left (10 x^{6}+24 x^{5}+15 x^{4}+60 c_1 \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{60 \left (x +1\right )^{2}} \\ y &= \frac {450^{{1}/{3}} {\left (\left (10 x^{6}+24 x^{5}+15 x^{4}+60 c_1 \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{60 \left (x +1\right )^{2}} \\ \end{align*}

✓ Mathematica. Time used: 4.188 (sec). Leaf size: 144

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

\begin{align*} y(x)&\to \frac {\sqrt [3]{\frac {10 x^6+24 x^5+15 x^4+60 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{15}}\\ y(x)&\to -\frac {\sqrt [3]{-\frac {1}{15}} \sqrt [3]{\frac {10 x^6+24 x^5+15 x^4+60 c_1}{(x+1)^2}}}{2^{2/3}}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{\frac {10 x^6+24 x^5+15 x^4+60 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{15}} \end{align*}

✓ Sympy. Time used: 2.118 (sec). Leaf size: 138

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

\[ \left [ y{\left (x \right )} = \frac {\sqrt [3]{\frac {C_{1} + 30 x^{6} + 72 x^{5} + 45 x^{4}}{x^{2} + 2 x + 1}} \left (- \sqrt [3]{150} - 3^{\frac {5}{6}} \sqrt [3]{50} i\right )}{60}, \ y{\left (x \right )} = \frac {\sqrt [3]{\frac {C_{1} + 30 x^{6} + 72 x^{5} + 45 x^{4}}{x^{2} + 2 x + 1}} \left (- \sqrt [3]{150} + 3^{\frac {5}{6}} \sqrt [3]{50} i\right )}{60}, \ y{\left (x \right )} = \frac {\sqrt [3]{150} \sqrt [3]{\frac {C_{1} + 30 x^{6} + 72 x^{5} + 45 x^{4}}{x^{2} + 2 x + 1}}}{30}\right ] \]