74.17.22 problem 22

Internal problem ID [16553]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 22
Date solved : Tuesday, January 28, 2025 at 09:10:41 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 38

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

\[ y = \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1-2 x +x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\left (3 x -3 x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 59

AsymptoticDSolveValue[x*(1-x)*D[y[x],{x,2}]+D[y[x],x]+2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (x^2-2 x+1\right )+c_2 \left (\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{3}-3 x^2+\left (x^2-2 x+1\right ) \log (x)+3 x\right ) \]