15.23.8 problem 8

Internal problem ID [3358]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 04:36:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 47

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)-3*(x^2+x)*diff(y(x),x)+(2+3*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {3}{2} x +\frac {9}{8} x^{2}+\frac {9}{16} x^{3}+\frac {27}{128} x^{4}+\frac {81}{1280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+\frac {3}{5} x +\frac {9}{35} x^{2}+\frac {3}{35} x^{3}+\frac {9}{385} x^{4}+\frac {27}{5005} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 88

ode=2*x^2*D[y[x],{x,2}]-3*(x+x^2)*D[y[x],x]+(2+3*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {27 x^5}{5005}+\frac {9 x^4}{385}+\frac {3 x^3}{35}+\frac {9 x^2}{35}+\frac {3 x}{5}+1\right ) x^2+c_2 \left (\frac {81 x^5}{1280}+\frac {27 x^4}{128}+\frac {9 x^3}{16}+\frac {9 x^2}{8}+\frac {3 x}{2}+1\right ) \sqrt {x} \]

✓ Sympy. Time used: 0.947 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + (3*x + 2)*y(x) - (3*x**2 + 3*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {3 x^{3}}{35} + \frac {9 x^{2}}{35} + \frac {3 x}{5} + 1\right ) + C_{1} \sqrt {x} \left (\frac {27 x^{4}}{128} + \frac {9 x^{3}}{16} + \frac {9 x^{2}}{8} + \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]