42.1.22 problem 3.48 (d)

Internal problem ID [6844]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.48 (d)
Date solved : Sunday, March 30, 2025 at 11:24:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.155 (sec). Leaf size: 312

ode=x*D[y[x],{x,2}]-2*D[y[x],x]+y[x]==Cos[x]; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (x^4 \left (\frac {\log (x)}{48}-\frac {5}{192}\right )-\frac {1}{12} x^3 \log (x)+\frac {x^2}{4}+\frac {x}{2}+1\right )+c_2 \left (-\frac {x^5}{806400}+\frac {x^4}{20160}-\frac {x^3}{720}+\frac {x^2}{40}-\frac {x}{4}+1\right ) x^3+\left (-\frac {x^5}{806400}+\frac {x^4}{20160}-\frac {x^3}{720}+\frac {x^2}{40}-\frac {x}{4}+1\right ) x^3 \left (\frac {x^6 \left (-20160 \log ^2(x)+141222 \log (x)-201569\right )}{3135283200}+\frac {x^5 (22277-114360 \log (x))}{435456000}+\frac {x^4 (69541-29064 \log (x))}{34836480}+\frac {x^3 (1860 \log (x)+193)}{388800}-\frac {1}{6 x^2}+\frac {x^2 (4 \log (x)-23)}{1152}-\frac {1}{6 x}+\frac {1}{36} x (-\log (x)-2)-\frac {\log (x)}{12}\right )+\left (\frac {x^6 (5791-672 \log (x))}{8709120}-\frac {589 x^5}{302400}-\frac {89 x^4}{8640}+\frac {19 x^3}{360}+\frac {x^2}{24}-\frac {x}{3}\right ) \left (x^4 \left (\frac {\log (x)}{48}-\frac {5}{192}\right )-\frac {1}{12} x^3 \log (x)+\frac {x^2}{4}+\frac {x}{2}+1\right ) \]

✗ Sympy

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

ValueError : ODE x*Derivative(y(x), (x, 2)) + y(x) - cos(x) - 2*Derivative(y(x), x) does not match hint 2nd_power_series_regular