73.10.8 problem 15.2 (h)
Internal
problem
ID
[15232]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
15.
General
solutions
to
Homogeneous
linear
differential
equations.
Additional
exercises
page
294
Problem
number
:
15.2
(h)
Date
solved
:
Monday, March 31, 2025 at 01:31:53 PM
CAS
classification
:
[[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\begin{align*} x y^{\prime \prime }-y^{\prime }+4 x^{3} y&=0 \end{align*}
With initial conditions
\begin{align*} y \left (\sqrt {\pi }\right )&=3\\ y^{\prime }\left (\sqrt {\pi }\right )&=4 \end{align*}
✓ Maple. Time used: 0.084 (sec). Leaf size: 24
ode:=x*diff(diff(y(x),x),x)-diff(y(x),x)+4*x^3*y(x) = 0;
ic:=y(Pi^(1/2)) = 3, D(y)(Pi^(1/2)) = 4;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \frac {-3 \cos \left (x^{2}\right ) \sqrt {\pi }-2 \sin \left (x^{2}\right )}{\sqrt {\pi }}
\]
✓ Mathematica. Time used: 0.019 (sec). Leaf size: 23
ode=x*D[y[x],{x,2}]-D[y[x],x]+4*x^3*y[x]==0;
ic={y[Sqrt[Pi]]==3,Derivative[1][y][Sqrt[Pi]]==4};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to -\frac {2 \sin \left (x^2\right )}{\sqrt {\pi }}-3 \cos \left (x^2\right )
\]
✓ Sympy. Time used: 0.304 (sec). Leaf size: 221
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**3*y(x) + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0)
ics = {y(sqrt(pi)): 3, Subs(Derivative(y(x), x), x, sqrt(pi)): 4}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = x \left (\frac {\left (- 3 Y_{\frac {1}{2}}\left (\pi \right ) - 3 \pi Y_{- \frac {1}{2}}\left (\pi \right ) + 3 \pi Y_{\frac {3}{2}}\left (\pi \right ) + 4 \sqrt {\pi } Y_{\frac {1}{2}}\left (\pi \right )\right ) J_{\frac {1}{2}}\left (x^{2}\right )}{\pi ^{\frac {3}{2}} J_{- \frac {1}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {3}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{- \frac {1}{2}}\left (\pi \right ) + \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{\frac {3}{2}}\left (\pi \right )} + \frac {\left (- 3 \pi J_{\frac {3}{2}}\left (\pi \right ) + 3 \pi J_{- \frac {1}{2}}\left (\pi \right ) - 4 \sqrt {\pi } J_{\frac {1}{2}}\left (\pi \right ) + 3 J_{\frac {1}{2}}\left (\pi \right )\right ) Y_{\frac {1}{2}}\left (x^{2}\right )}{\pi ^{\frac {3}{2}} J_{- \frac {1}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {3}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{- \frac {1}{2}}\left (\pi \right ) + \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{\frac {3}{2}}\left (\pi \right )}\right )
\]