74.18.63 problem 69

Internal problem ID [16541]
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 : 69
Date solved : Thursday, March 13, 2025 at 08:17:28 AM
CAS classification : [_Jacobi]

\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: 0.052 (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);
 

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

✓ Mathematica. Time used: 0.009 (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: 0.954 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (2*x + 1)*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)
 

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