50.19.10 problem 3(a)

Internal problem ID [8118]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 3(a)
Date solved : Monday, January 27, 2025 at 03:44:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.152 (sec). Leaf size: 37

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

\[ y = c_{1} x^{2} \left (1-\frac {2}{9} x^{2}+\frac {26}{675} x^{4}-\frac {1742}{297675} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {1}{3} x^{2}+\frac {17}{270} x^{4}-\frac {173}{17010} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 74

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

\[ y(x)\to c_2 \left (\frac {32351 x^8}{40186125}-\frac {1742 x^6}{297675}+\frac {26 x^4}{675}-\frac {2 x^2}{9}+1\right ) x^2+c_1 \left (\frac {10471 x^8}{7144200}-\frac {173 x^6}{17010}+\frac {17 x^4}{270}-\frac {x^2}{3}+1\right ) x \]