49.17.7 problem 1(g)

Internal problem ID [7714]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 1(g)
Date solved : Monday, January 27, 2025 at 03:10:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\cos \left (x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 39

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

\[ y = c_{1} x^{-i} \left (1+\left (\frac {1}{12}+\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}+\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}-\frac {17 i}{4838400}\right ) x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (\frac {1}{12}-\frac {i}{24}\right ) x^{2}+\left (\frac {29}{28800}-\frac {67 i}{28800}\right ) x^{4}+\left (-\frac {893}{14515200}+\frac {17 i}{4838400}\right ) x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.057 (sec). Leaf size: 112

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+Sin[x]*D[y[x],x]+Cos[x]*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 x^{-i} \left (\left (-\frac {26459}{59222016000}-\frac {12449 i}{7402752000}\right ) x^8-\left (\frac {893}{14515200}+\frac {17 i}{4838400}\right ) x^6+\left (\frac {29}{28800}+\frac {67 i}{28800}\right ) x^4+\left (\frac {1}{12}+\frac {i}{24}\right ) x^2+1\right )+c_2 x^i \left (\left (-\frac {26459}{59222016000}+\frac {12449 i}{7402752000}\right ) x^8-\left (\frac {893}{14515200}-\frac {17 i}{4838400}\right ) x^6+\left (\frac {29}{28800}-\frac {67 i}{28800}\right ) x^4+\left (\frac {1}{12}-\frac {i}{24}\right ) x^2+1\right ) \]