6.15.23 problem 19

Internal problem ID [2021]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 05:22:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (3-2 x \right ) y^{\prime }+\left (4+3 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*(3-2*x)*diff(y(x),x)+(4+3*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{2} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-7 x +\frac {63}{4} x^{2}-\frac {77}{4} x^{3}+\frac {1001}{64} x^{4}-\frac {3003}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12 x -\frac {157}{4} x^{2}+\frac {2063}{36} x^{3}-\frac {59875}{1152} x^{4}+\frac {323399}{9600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 122

ode=x^2*D[y[x],{x,2}]-x*(3-2*x)*D[y[x],x]+(4+3*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {3003 x^5}{320}+\frac {1001 x^4}{64}-\frac {77 x^3}{4}+\frac {63 x^2}{4}-7 x+1\right ) x^2+c_2 \left (\left (\frac {323399 x^5}{9600}-\frac {59875 x^4}{1152}+\frac {2063 x^3}{36}-\frac {157 x^2}{4}+12 x\right ) x^2+\left (-\frac {3003 x^5}{320}+\frac {1001 x^4}{64}-\frac {77 x^3}{4}+\frac {63 x^2}{4}-7 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.309 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(3 - 2*x)*Derivative(y(x), x) + (3*x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{2} \left (- \frac {77 x^{3}}{4} + \frac {63 x^{2}}{4} - 7 x + 1\right ) + O\left (x^{6}\right ) \]