74.17.12 problem 12

Internal problem ID [16543]
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 : 12
Date solved : Tuesday, January 28, 2025 at 09:10:30 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+\left (\frac {16}{3 x}-1\right ) y^{\prime }-\frac {16 y}{3 x^{2}}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 45

Order:=6; 
dsolve(diff(y(x),x$2)+(16/3*1/x-1)*diff(y(x),x)-16/3*1/x^2*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{16}/{3}}}+c_{2} x \left (1+\frac {3}{22} x +\frac {9}{550} x^{2}+\frac {27}{15400} x^{3}+\frac {81}{477400} x^{4}+\frac {243}{16231600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 82

AsymptoticDSolveValue[D[y[x],{x,2}]+(16/3*1/x-1)*D[y[x],x]-16/3*1/x^2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 x \left (\frac {243 x^5}{16231600}+\frac {81 x^4}{477400}+\frac {27 x^3}{15400}+\frac {9 x^2}{550}+\frac {3 x}{22}+1\right )+\frac {c_2 \left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )}{x^{16/3}} \]