54.3.20 problem 20

Internal problem ID [8576]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 20
Date solved : Monday, January 27, 2025 at 04:16:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 y^{\prime \prime }+9 x y^{\prime }-36 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 44

Order:=8; 
dsolve(2*diff(y(x),x$2)+9*x*diff(y(x),x)-36*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (\frac {27}{4} x^{4}+9 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {9}{4} x^{3}+\frac {81}{160} x^{5}-\frac {243}{4480} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 47

AsymptoticDSolveValue[2*D[y[x],{x,2}]+9*x*D[y[x],x]-36*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {27 x^4}{4}+9 x^2+1\right )+c_2 \left (-\frac {243 x^7}{4480}+\frac {81 x^5}{160}+\frac {9 x^3}{4}+x\right ) \]