87.15.21 problem 21

Internal problem ID [23569]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 115
Problem number : 21
Date solved : Thursday, October 02, 2025 at 09:43:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 77

Order:=6; 
ode:=(1+a*cos(2*x))*diff(diff(y(x),x),x)+lambda*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {\lambda \,x^{2}}{2 \left (a +1\right )}-\frac {\lambda \left (-\lambda +4 a \right ) x^{4}}{24 \left (a +1\right )^{2}}\right ) y \left (0\right )+\left (x -\frac {\lambda \,x^{3}}{6 \left (a +1\right )}-\frac {\lambda \left (-\lambda +12 a \right ) x^{5}}{120 \left (a +1\right )^{2}}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 98

ode=(1+a*Cos[2*x])*D[y[x],{x,2}]+\[Lambda]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {\lambda ^2 x^5}{120 (a+1)^2}-\frac {a \lambda x^5}{10 (a+1)^2}-\frac {\lambda x^3}{6 (a+1)}+x\right )+c_1 \left (\frac {\lambda ^2 x^4}{24 (a+1)^2}-\frac {a \lambda x^4}{6 (a+1)^2}-\frac {\lambda x^2}{2 (a+1)}+1\right ) \]

✓ Sympy. Time used: 0.549 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(lambda_*y(x) + (a*cos(2*x) + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\lambda _{}^{2} x^{4}}{24 \left (a^{2} \cos ^{2}{\left (2 x \right )} + 2 a \cos {\left (2 x \right )} + 1\right )} - \frac {\lambda _{} x^{2}}{2 \left (a \cos {\left (2 x \right )} + 1\right )} + 1\right ) + C_{1} x \left (- \frac {\lambda _{} x^{2}}{6 \left (a \cos {\left (2 x \right )} + 1\right )} + 1\right ) + O\left (x^{6}\right ) \]