12.13.26 problem 29

Internal problem ID [1917]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 29
Date solved : Saturday, March 29, 2025 at 11:43:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\beta \,x^{2}+\alpha x +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 265

Order:=6; 
ode:=(beta*x^2+alpha*x+1)*diff(diff(y(x),x),x)+(delta*x+gamma)*diff(y(x),x)+epsilon*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {\epsilon \,x^{2}}{2}+\frac {\epsilon \left (\alpha +\gamma \right ) x^{3}}{6}+\frac {\epsilon \left (-\alpha ^{2}-\frac {3}{2} \alpha \gamma -\frac {1}{2} \gamma ^{2}+\beta +\delta +\frac {1}{2} \epsilon \right ) x^{4}}{12}-\frac {\left (\frac {\left (\alpha +\frac {\gamma }{2}\right ) \epsilon }{3}-\frac {\gamma ^{3}}{12}-\frac {\alpha \,\gamma ^{2}}{2}+\frac {\left (-\frac {11 \alpha ^{2}}{4}+2 \beta +\frac {5 \delta }{4}\right ) \gamma }{3}+\alpha \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {3 \delta }{4}\right )\right ) \epsilon \,x^{5}}{10}\right ) y \left (0\right )+\left (x -\frac {\gamma \,x^{2}}{2}+\frac {\left (\alpha \gamma +\gamma ^{2}-\delta -\epsilon \right ) x^{3}}{6}+\frac {\left (\left (2 \alpha +2 \gamma \right ) \epsilon -\gamma ^{3}-3 \alpha \,\gamma ^{2}+\left (-2 \alpha ^{2}+2 \beta +3 \delta \right ) \gamma +2 \delta \alpha \right ) x^{4}}{24}+\frac {\left (\epsilon ^{2}+\left (-6 \alpha ^{2}-9 \alpha \gamma -3 \gamma ^{2}+6 \beta +4 \delta \right ) \epsilon +\gamma ^{4}+6 \alpha \,\gamma ^{3}+\left (11 \alpha ^{2}-8 \beta -6 \delta \right ) \gamma ^{2}-12 \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {7 \delta }{6}\right ) \alpha \gamma +6 \delta \left (-\alpha ^{2}+\beta +\frac {\delta }{2}\right )\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 561

ode=(1+\[Alpha]*x+\[Beta]*x^2)*D[y[x],{x,2}]+(\[Gamma]+\[Delta]*x)*D[y[x],x]+\[Epsilon]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 1.878 (sec). Leaf size: 201

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
Gamma = symbols("Gamma") 
delta = symbols("delta") 
epsilon = symbols("epsilon") 
y = Function("y") 
ode = Eq(epsilon*y(x) + (Gamma + delta*x)*Derivative(y(x), x) + (Alpha*x + BETA*x**2 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {\mathrm {A}^{2} \epsilon x^{4}}{12} - \frac {\mathrm {A} \Gamma \epsilon x^{4}}{8} + \frac {\mathrm {A} \epsilon x^{3}}{6} + \frac {\beta \epsilon x^{4}}{12} - \frac {\Gamma ^{2} \epsilon x^{4}}{24} + \frac {\Gamma \epsilon x^{3}}{6} + \frac {\delta \epsilon x^{4}}{12} + \frac {\epsilon ^{2} x^{4}}{24} - \frac {\epsilon x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {\mathrm {A}^{2} \Gamma x^{3}}{12} - \frac {\mathrm {A} \Gamma ^{2} x^{3}}{8} + \frac {\mathrm {A} \Gamma x^{2}}{6} + \frac {\mathrm {A} \delta x^{3}}{12} + \frac {\mathrm {A} \epsilon x^{3}}{12} + \frac {\beta \Gamma x^{3}}{12} - \frac {\Gamma ^{3} x^{3}}{24} + \frac {\Gamma ^{2} x^{2}}{6} + \frac {\Gamma \delta x^{3}}{8} + \frac {\Gamma \epsilon x^{3}}{12} - \frac {\Gamma x}{2} - \frac {\delta x^{2}}{6} - \frac {\epsilon x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]