52.2.19 problem 19

Internal problem ID [8270]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 19
Date solved : Monday, January 27, 2025 at 03:47:32 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 3 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: 0.013 (sec). Leaf size: 52

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

\[ y = c_{1} x^{{1}/{3}} \left (1+\frac {1}{3} x +\frac {1}{18} x^{2}+\frac {1}{162} x^{3}+\frac {1}{1944} x^{4}+\frac {1}{29160} x^{5}+\frac {1}{524880} x^{6}+\frac {1}{11022480} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{2} x +\frac {1}{10} x^{2}+\frac {1}{80} x^{3}+\frac {1}{880} x^{4}+\frac {1}{12320} x^{5}+\frac {1}{209440} x^{6}+\frac {1}{4188800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

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

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {x^7}{11022480}+\frac {x^6}{524880}+\frac {x^5}{29160}+\frac {x^4}{1944}+\frac {x^3}{162}+\frac {x^2}{18}+\frac {x}{3}+1\right )+c_2 \left (\frac {x^7}{4188800}+\frac {x^6}{209440}+\frac {x^5}{12320}+\frac {x^4}{880}+\frac {x^3}{80}+\frac {x^2}{10}+\frac {x}{2}+1\right ) \]