85.76.3 problem 4 (a)

Internal problem ID [22981]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. B Exercises at page 330
Problem number : 4 (a)
Date solved : Thursday, October 02, 2025 at 09:17:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-x \right ) y^{\prime \prime }+\left (2-4 x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 59

Order:=6; 
ode:=(1-x)*diff(diff(y(x),x),x)+(2-4*x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {3}{8} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {7}{6} x^{3}-\frac {3}{4} x^{4}+\frac {73}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 68

ode=(1-x)*D[y[x],{x,2}]+(2-4*x)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{30}+\frac {3 x^4}{8}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {73 x^5}{120}-\frac {3 x^4}{4}+\frac {7 x^3}{6}-x^2+x\right ) \]

✓ Sympy. Time used: 0.320 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - x)*Derivative(y(x), (x, 2)) + (2 - 4*x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {3 x^{4}}{8} - \frac {x^{3}}{6} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {3 x^{3}}{4} + \frac {7 x^{2}}{6} - x + 1\right ) + O\left (x^{6}\right ) \]