87.14.14 problem 14

Internal problem ID [23533]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:42:46 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }-\left (1+\frac {3}{2 x}\right ) y^{\prime }+\frac {3 y}{2 x^{2}}&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{{3}/{2}} {\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x)-(1+3/2/x)*diff(y(x),x)+3/2/x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{{3}/{2}} {\mathrm e}^{x}+c_2 \left ({\mathrm e}^{x} \sqrt {\pi }\, \left (-1+\operatorname {erf}\left (\sqrt {x}\right )\right ) x^{{3}/{2}}+x \right ) \]

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 39

ode=D[y[x],{x,2}]-(1+3/(2*x))*D[y[x],x]+(3/(2*x^2))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 e^x x^{3/2}+2 c_1 e^x x^{3/2} \Gamma \left (\frac {1}{2},x\right )-2 c_1 x \end{align*}

✓ Sympy. Time used: 0.802 (sec). Leaf size: 542

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(1 + 3/(2*x))*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 3*y(x)/(2*x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]