54.3.19 problem 19

Internal problem ID [8575]
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 : 19
Date solved : Monday, January 27, 2025 at 04:16:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 x y^{\prime }+7 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: 49

Order:=8; 
dsolve(diff(y(x),x$2)+3*x*diff(y(x),x)+7*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {7}{2} x^{2}+\frac {91}{24} x^{4}-\frac {1729}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {5}{3} x^{3}+\frac {4}{3} x^{5}-\frac {44}{63} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 56

AsymptoticDSolveValue[D[y[x],{x,2}]+3*x*D[y[x],x]+7*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (-\frac {44 x^7}{63}+\frac {4 x^5}{3}-\frac {5 x^3}{3}+x\right )+c_1 \left (-\frac {1729 x^6}{720}+\frac {91 x^4}{24}-\frac {7 x^2}{2}+1\right ) \]