52.2.13 problem 13

Internal problem ID [8264]
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 : 13
Date solved : Monday, January 27, 2025 at 03:47:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 44

Order:=8; 
dsolve(x^2*diff(y(x),x$2)+(5/3*x+x^2)*diff(y(x),x)-1/3*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{2} x^{{4}/{3}} \left (1-\frac {1}{7} x +\frac {1}{35} x^{2}-\frac {1}{195} x^{3}+\frac {1}{1248} x^{4}-\frac {1}{9120} x^{5}+\frac {1}{75240} x^{6}-\frac {1}{693000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1-3 x +\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 72

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+(5/3*x+x^2)*D[y[x],x]-1/3*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {x^7}{693000}+\frac {x^6}{75240}-\frac {x^5}{9120}+\frac {x^4}{1248}-\frac {x^3}{195}+\frac {x^2}{35}-\frac {x}{7}+1\right )+\frac {c_2 (1-3 x)}{x} \]