50.19.15 problem 4(b)

Internal problem ID [8123]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 4(b)
Date solved : Monday, January 27, 2025 at 03:44:15 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x y^{\prime \prime }+\left (3-x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 52

Order:=8; 
dsolve(2*x*diff(y(x),x$2)+(3-x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (1+\frac {1}{3} x +\frac {1}{15} x^{2}+\frac {1}{105} x^{3}+\frac {1}{945} x^{4}+\frac {1}{10395} x^{5}+\frac {1}{135135} x^{6}+\frac {1}{2027025} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 113

AsymptoticDSolveValue[2*x*D[y[x],{x,2}]+(3-x)*D[y[x],x]-y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^7}{2027025}+\frac {x^6}{135135}+\frac {x^5}{10395}+\frac {x^4}{945}+\frac {x^3}{105}+\frac {x^2}{15}+\frac {x}{3}+1\right )+\frac {c_2 \left (\frac {x^7}{645120}+\frac {x^6}{46080}+\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right )}{\sqrt {x}} \]