6.3.12 problem 13
Internal
problem
ID
[1589]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
separable
equations.
Section
2.2
Page
52
Problem
number
:
13
Date
solved
:
Tuesday, September 30, 2025 at 04:37:40 AM
CAS
classification
:
[_separable]
\begin{align*} \left (3 y^{2}+4 y\right ) y^{\prime }+2 x +\cos \left (x \right )&=0 \end{align*}
With initial conditions
\begin{align*}
y \left (0\right )&=1 \\
\end{align*}
✓ Maple. Time used: 0.436 (sec). Leaf size: 102
ode:=(3*y(x)^2+4*y(x))*diff(y(x),x)+2*x+cos(x) = 0;
ic:=[y(0) = 1];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = \frac {\left (260-108 x^{2}-108 \sin \left (x \right )+12 \sqrt {441-390 x^{2}-390 \sin \left (x \right )+81 x^{4}+162 x^{2} \sin \left (x \right )+81 \sin \left (x \right )^{2}}\right )^{{1}/{3}}}{6}+\frac {8}{3 \left (260-108 x^{2}-108 \sin \left (x \right )+12 \sqrt {441-390 x^{2}-390 \sin \left (x \right )+81 x^{4}+162 x^{2} \sin \left (x \right )+81 \sin \left (x \right )^{2}}\right )^{{1}/{3}}}-\frac {2}{3}
\]
✓ Mathematica. Time used: 1.653 (sec). Leaf size: 127
ode=(3*y[x]^2+4*y[x])*D[y[x],x]+2*x+Cos[x]==0;
ic=y[0]==1;
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2^{2/3} \left (-27 x^2+\sqrt {\left (27 x^2+27 \sin (x)-65\right )^2-256}-27 \sin (x)+65\right )^{2/3}-4 \sqrt [3]{-27 x^2+\sqrt {\left (27 x^2+27 \sin (x)-65\right )^2-256}-27 \sin (x)+65}+8 \sqrt [3]{2}}{6 \sqrt [3]{-27 x^2+\sqrt {\left (27 x^2+27 \sin (x)-65\right )^2-256}-27 \sin (x)+65}} \end{align*}
✓ Sympy. Time used: 91.490 (sec). Leaf size: 221
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x + (3*y(x)**2 + 4*y(x))*Derivative(y(x), x) + cos(x),0)
ics = {y(0): 1}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\sqrt [3]{\frac {27 x^{2}}{2} + \frac {3 \sqrt {3} \sqrt {27 x^{4} + 54 x^{2} \sin {\left (x \right )} - 130 x^{2} + 27 \sin ^{2}{\left (x \right )} - 130 \sin {\left (x \right )} + 147}}{2} + \frac {27 \sin {\left (x \right )}}{2} - \frac {65}{2}}}{6} - \frac {\sqrt {3} i \sqrt [3]{\frac {27 x^{2}}{2} + \frac {3 \sqrt {3} \sqrt {27 x^{4} + 54 x^{2} \sin {\left (x \right )} - 130 x^{2} + 27 \sin ^{2}{\left (x \right )} - 130 \sin {\left (x \right )} + 147}}{2} + \frac {27 \sin {\left (x \right )}}{2} - \frac {65}{2}}}{6} - \frac {2}{3} - \frac {8}{3 \left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {27 x^{2}}{2} + \frac {3 \sqrt {3} \sqrt {27 x^{4} + 54 x^{2} \sin {\left (x \right )} - 130 x^{2} + 27 \sin ^{2}{\left (x \right )} - 130 \sin {\left (x \right )} + 147}}{2} + \frac {27 \sin {\left (x \right )}}{2} - \frac {65}{2}}}
\]