74.18.64 problem 70

Internal problem ID [16550]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 70
Date solved : Monday, March 31, 2025 at 02:56:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (1+x \right ) y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }-10 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 44

Order:=6; 
ode:=x*(1+x)*diff(diff(y(x),x),x)+(-2*x+1)*diff(y(x),x)-10*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (-17\right ) x -\frac {157}{2} x^{2}-\frac {404}{3} x^{3}-\frac {625}{6} x^{4}-\frac {162}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+10 x +30 x^{2}+40 x^{3}+25 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 95

ode=x*(1+x)*D[y[x],{x,2}]+(1-2*x)*D[y[x],x]-10*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (6 x^5+25 x^4+40 x^3+30 x^2+10 x+1\right )+c_2 \left (-\frac {162 x^5}{5}-\frac {625 x^4}{6}-\frac {404 x^3}{3}-\frac {157 x^2}{2}+\left (6 x^5+25 x^4+40 x^3+30 x^2+10 x+1\right ) \log (x)-17 x\right ) \]

✓ Sympy. Time used: 0.997 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 1)*Derivative(y(x), (x, 2)) + (1 - 2*x)*Derivative(y(x), x) - 10*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {125 x^{5}}{18} + \frac {625 x^{4}}{36} + \frac {250 x^{3}}{9} + 25 x^{2} + 10 x + 1\right ) + O\left (x^{6}\right ) \]