83.16.4 problem 2 (b)
Internal
problem
ID
[22045]
Book
:
Differential
Equations
By
Kaj
L.
Nielsen.
Second
edition
1966.
Barnes
and
nobel.
66-28306
Section
:
Examination
I.
page
253
Problem
number
:
2
(b)
Date
solved
:
Thursday, October 02, 2025 at 08:23:01 PM
CAS
classification
:
[_linear]
\begin{align*} \left (x^{3}+3\right ) y^{\prime }+2 y x +5 x^{2}&=0 \end{align*}
With initial conditions
\begin{align*}
y \left (2\right )&=1 \\
\end{align*}
✓ Maple. Time used: 0.406 (sec). Leaf size: 170
ode:=(x^3+3)*diff(y(x),x)+2*x*y(x)+5*x^2 = 0;
ic:=[y(2) = 1];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = \left (x +3^{{1}/{3}}\right )^{\frac {2 \,3^{{2}/{3}}}{9}} \left (x^{2}-3^{{1}/{3}} x +3^{{2}/{3}}\right )^{-\frac {3^{{2}/{3}}}{9}} {\mathrm e}^{-\frac {2 \,3^{{1}/{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{{2}/{3}} x -3\right )}{9}\right )}{3}} \left (-5 \int _{2}^{x}\frac {\textit {\_z1}^{2} \left (\textit {\_z1} +3^{{1}/{3}}\right )^{-\frac {2 \,3^{{2}/{3}}}{9}} \left (\textit {\_z1}^{2}-3^{{1}/{3}} \textit {\_z1} +3^{{2}/{3}}\right )^{\frac {3^{{2}/{3}}}{9}} {\mathrm e}^{\frac {2 \,3^{{1}/{6}} \arctan \left (\frac {2 \,3^{{1}/{6}} \textit {\_z1}}{3}-\frac {\sqrt {3}}{3}\right )}{3}}}{\textit {\_z1}^{3}+3}d \textit {\_z1} +\left (2+3^{{1}/{3}}\right )^{-\frac {2 \,3^{{2}/{3}}}{9}} \left (4-2 \,3^{{1}/{3}}+3^{{2}/{3}}\right )^{\frac {3^{{2}/{3}}}{9}} {\mathrm e}^{\frac {2 \,3^{{1}/{6}} \arctan \left (\frac {4 \,3^{{1}/{6}}}{3}-\frac {\sqrt {3}}{3}\right )}{3}}\right )
\]
✓ Mathematica. Time used: 0.446 (sec). Leaf size: 253
ode=(x^3+3)*D[y[x],x]+2*x*y[x]+5*x^2==0;
ic={y[2]==1};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \exp \left (\frac {6 \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-6 \arctan \left (\frac {\sqrt [3]{3}-4}{3^{5/6}}\right )+\sqrt {3} \left (2 \log \left (3^{2/3} x+3\right )-\log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )\right )}{3\ 3^{5/6}}\right ) \left (e^{\frac {2 \arctan \left (\frac {\sqrt [3]{3}-4}{3^{5/6}}\right )}{3^{5/6}}} \int _2^x-\frac {5 \exp \left (\frac {1}{9} \left (3^{2/3} \left (\log \left (\sqrt [3]{3} K[1]^2-3^{2/3} K[1]+3\right )-2 \log \left (3^{2/3} K[1]+3\right )\right )-6 \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 K[1]}{3^{5/6}}\right )\right )\right ) K[1]^2}{K[1]^3+3}dK[1]+\exp \left (\frac {\log \left (3+4 \sqrt [3]{3}-2\ 3^{2/3}\right )-2 \log \left (3+2\ 3^{2/3}\right )}{3 \sqrt [3]{3}}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(5*x**2 + 2*x*y(x) + (x**3 + 3)*Derivative(y(x), x),0)
ics = {y(2): 1}
dsolve(ode,func=y(x),ics=ics)
Timed Out