68.16.13 problem 13

Internal problem ID [17806]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 13
Date solved : Thursday, October 02, 2025 at 02:28:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 39

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

\[ y = \left (1+\frac {3}{4} x^{2}-\frac {5}{32} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{3}-\frac {1}{32} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (-\frac {x^5}{32}+\frac {x^3}{12}+x\right )+c_1 \left (-\frac {5 x^4}{32}+\frac {3 x^2}{4}+1\right ) \]

✓ Sympy. Time used: 0.277 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {5 x^{4}}{32} + \frac {3 x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {x^{2}}{12} + 1\right ) + O\left (x^{6}\right ) \]