36.3.16 problem 17
Internal
problem
ID
[6337]
Book
:
Fundamentals
of
Differential
Equations.
By
Nagle,
Saff
and
Snider.
9th
edition.
Boston.
Pearson
2018.
Section
:
Chapter
2,
First
order
differential
equations.
Section
2.4,
Exact
equations.
Exercises.
page
64
Problem
number
:
17
Date
solved
:
Sunday, March 30, 2025 at 10:52:01 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} \frac {1}{y}-\left (3 y-\frac {x}{y^{2}}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 18
ode:=1/y(x)-(3*y(x)-x/y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
-\frac {c_1}{y}+x -\frac {3 y^{3}}{4} = 0
\]
✓ Mathematica. Time used: 36.497 (sec). Leaf size: 870
ode=1/y[x]-(3*y[x]-x/y[x]^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt {-\frac {2 \sqrt {6} x}{\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}-\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}-\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}}{\sqrt {6}} \\
y(x)\to \frac {\sqrt {-\frac {2 \sqrt {6} x}{\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}-\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}-\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}-\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}{\sqrt {6}} \\
y(x)\to \frac {\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}-\sqrt {\frac {2 \sqrt {6} x}{\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}-\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}-\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}}{\sqrt {6}} \\
y(x)\to \frac {\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt {\frac {2 \sqrt {6} x}{\sqrt {\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}+\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}-\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}-\frac {4 c_1}{\sqrt [3]{3 x^2-\sqrt {9 x^4-64 c_1{}^3}}}}}{\sqrt {6}} \\
\end{align*}
✓ Sympy. Time used: 22.458 (sec). Leaf size: 682
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x/y(x)**2 - 3*y(x))*Derivative(y(x), x) + 1/y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt {6} \left (- \sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} - \sqrt {\frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} - \frac {2 \sqrt {6} x}{\sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}} - \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}\right )}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \left (- \sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt {\frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} - \frac {2 \sqrt {6} x}{\sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}} - \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}\right )}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \left (\sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} - \sqrt {\frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \frac {2 \sqrt {6} x}{\sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}} - \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}\right )}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \left (\sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt {\frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \frac {2 \sqrt {6} x}{\sqrt {- \frac {4 C_{1}}{\sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}} + \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}} - \sqrt [3]{3 x^{2} + \sqrt {64 C_{1}^{3} + 9 x^{4}}}}\right )}{6}\right ]
\]