67.10.8 problem 15.2 (h)

Internal problem ID [16590]
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 : Thursday, October 02, 2025 at 01:36:43 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,op(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: 0.013 (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]
 

\begin{align*} y(x)&\to -\frac {2 \sin \left (x^2\right )}{\sqrt {\pi }}-3 \cos \left (x^2\right ) \end{align*}

✓ Sympy. Time used: 0.185 (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)
 

\[ y{\left (x \right )} = x \left (\frac {\left (- 3 Y_{\frac {1}{2}}\left (\pi \right ) - 3 \pi Y_{- \frac {1}{2}}\left (\pi \right ) + 3 \pi Y_{\frac {3}{2}}\left (\pi \right ) + 4 \sqrt {\pi } Y_{\frac {1}{2}}\left (\pi \right )\right ) J_{\frac {1}{2}}\left (x^{2}\right )}{\pi ^{\frac {3}{2}} J_{- \frac {1}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {3}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{- \frac {1}{2}}\left (\pi \right ) + \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{\frac {3}{2}}\left (\pi \right )} + \frac {\left (- 3 \pi J_{\frac {3}{2}}\left (\pi \right ) + 3 \pi J_{- \frac {1}{2}}\left (\pi \right ) - 4 \sqrt {\pi } J_{\frac {1}{2}}\left (\pi \right ) + 3 J_{\frac {1}{2}}\left (\pi \right )\right ) Y_{\frac {1}{2}}\left (x^{2}\right )}{\pi ^{\frac {3}{2}} J_{- \frac {1}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {3}{2}}\left (\pi \right ) Y_{\frac {1}{2}}\left (\pi \right ) - \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{- \frac {1}{2}}\left (\pi \right ) + \pi ^{\frac {3}{2}} J_{\frac {1}{2}}\left (\pi \right ) Y_{\frac {3}{2}}\left (\pi \right )}\right ) \]