58.4.7 problem 7
Internal
problem
ID
[14577]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
2,
section
2.2
(Separable
equations).
Exercises
page
47
Problem
number
:
7
Date
solved
:
Thursday, October 02, 2025 at 09:42:37 AM
CAS
classification
:
[_separable]
\begin{align*} \left (4+x \right ) \left (1+y^{2}\right )+y \left (x^{2}+3 x +2\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 114
ode:=(x+4)*(1+y(x)^2)+y(x)*(x^2+3*x+2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {-x^{6}-6 x^{5}+c_1 \,x^{4}+\left (8 c_1 +100\right ) x^{3}+\left (24 c_1 +345\right ) x^{2}+\left (32 c_1 +474\right ) x +16 c_1 +239}}{\left (x +1\right )^{3}} \\
y &= -\frac {\sqrt {-x^{6}-6 x^{5}+c_1 \,x^{4}+\left (8 c_1 +100\right ) x^{3}+\left (24 c_1 +345\right ) x^{2}+\left (32 c_1 +474\right ) x +16 c_1 +239}}{\left (x +1\right )^{3}} \\
\end{align*}
✓ Mathematica. Time used: 4.594 (sec). Leaf size: 126
ode=(x+4)*(y[x]^2+1) + y[x]*(x^2+3*x+2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt {-(x+1)^6+e^{2 c_1} (x+2)^4}}{(x+1)^3}\\ y(x)&\to \frac {\sqrt {-(x+1)^6+e^{2 c_1} (x+2)^4}}{(x+1)^3}\\ y(x)&\to -i\\ y(x)&\to i\\ y(x)&\to \frac {(x+1)^3}{\sqrt {-(x+1)^6}}\\ y(x)&\to \frac {\sqrt {-(x+1)^6}}{(x+1)^3} \end{align*}
✓ Sympy. Time used: 8.416 (sec). Leaf size: 393
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x + 4)*(y(x)**2 + 1) + (x**2 + 3*x + 2)*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {\frac {x^{4} e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} + \frac {8 x^{3} e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} + \frac {24 x^{2} e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} + \frac {32 x e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} - 1 + \frac {16 e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1}}, \ y{\left (x \right )} = \sqrt {\frac {x^{4} e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} + \frac {8 x^{3} e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} + \frac {24 x^{2} e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} + \frac {32 x e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1} - 1 + \frac {16 e^{2 C_{1}}}{x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x + 1}}\right ]
\]