54.3.12 problem 12

Internal problem ID [8568]
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 : 12
Date solved : Monday, January 27, 2025 at 04:16:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }+3 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: 39

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

\[ y = \left (1-\frac {3}{2} x^{2}-\frac {3}{8} x^{4}+\frac {7}{80} x^{6}\right ) y \left (0\right )+\left (\frac {1}{3} x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {x^3}{3}+x\right )+c_1 \left (\frac {7 x^6}{80}-\frac {3 x^4}{8}-\frac {3 x^2}{2}+1\right ) \]