15.7.14 problem 14
Internal
problem
ID
[2995]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
11,
page
45
Problem
number
:
14
Date
solved
:
Sunday, March 30, 2025 at 01:04:10 AM
CAS
classification
:
[_rational, _Bernoulli]
\begin{align*} 3 y^{\prime }+\frac {2 y}{x +1}&=\frac {x}{y^{2}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 135
ode:=3*diff(y(x),x)+2*y(x)/(1+x) = x/y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {18^{{1}/{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_1 \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}}}{6 \left (x +1\right )^{2}} \\
y &= -\frac {18^{{1}/{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_1 \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{12 \left (x +1\right )^{2}} \\
y &= \frac {18^{{1}/{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_1 \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12 \left (x +1\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 4.283 (sec). Leaf size: 144
ode=3*D[y[x],x]+2*y[x]/(x+1)==x/y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{\frac {3 x^4+8 x^3+6 x^2+12 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{3}} \\
y(x)\to -\frac {\sqrt [3]{-\frac {1}{3}} \sqrt [3]{\frac {3 x^4+8 x^3+6 x^2+12 c_1}{(x+1)^2}}}{2^{2/3}} \\
y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\frac {3 x^4+8 x^3+6 x^2+12 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{3}} \\
\end{align*}
✓ Sympy. Time used: 3.084 (sec). Leaf size: 138
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x/y(x)**2 + 3*Derivative(y(x), x) + 2*y(x)/(x + 1),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt [3]{\frac {C_{1} + 9 x^{4} + 24 x^{3} + 18 x^{2}}{x^{2} + 2 x + 1}} \left (- \sqrt [3]{6} - \sqrt [3]{2} \cdot 3^{\frac {5}{6}} i\right )}{12}, \ y{\left (x \right )} = \frac {\sqrt [3]{\frac {C_{1} + 9 x^{4} + 24 x^{3} + 18 x^{2}}{x^{2} + 2 x + 1}} \left (- \sqrt [3]{6} + \sqrt [3]{2} \cdot 3^{\frac {5}{6}} i\right )}{12}, \ y{\left (x \right )} = \frac {\sqrt [3]{6} \sqrt [3]{\frac {C_{1} + 9 x^{4} + 24 x^{3} + 18 x^{2}}{x^{2} + 2 x + 1}}}{6}\right ]
\]