54.9.25 problem 26

Internal problem ID [8695]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 26
Date solved : Monday, January 27, 2025 at 04:19:27 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 52

Order:=8; 
dsolve(x*diff(y(x),x$2)+(2-x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} \left (1+\frac {1}{2} x +\frac {1}{6} x^{2}+\frac {1}{24} x^{3}+\frac {1}{120} x^{4}+\frac {1}{720} x^{5}+\frac {1}{5040} x^{6}+\frac {1}{40320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.076 (sec). Leaf size: 90

AsymptoticDSolveValue[x*D[y[x],{x,2}]+(2-x)*D[y[x],x]-y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{720}+\frac {x^4}{120}+\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^6}{5040}+\frac {x^5}{720}+\frac {x^4}{120}+\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+1\right ) \]