75.25.4 problem 760

Internal problem ID [17156]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.3. Finding periodic solutions of linear differential equations. Exercises page 187
Problem number : 760
Date solved : Monday, March 31, 2025 at 03:43:29 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=\arcsin \left (\sin \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = arcsin(sin(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 x} \left (c_2 +c_1 x +\int {\mathrm e}^{-2 x} \arcsin \left (\sin \left (x \right )\right )d x x -\int {\mathrm e}^{-2 x} x \arcsin \left (\sin \left (x \right )\right )d x \right ) \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 59

ode=D[y[x],{x,2}]-4*D[y[x],x]+4*y[x]==ArcSin[Sin[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{2 x} \left (x \int _1^xe^{-2 K[2]} \arcsin (\sin (K[2]))dK[2]+\int _1^x-e^{-2 K[1]} \arcsin (\sin (K[1])) K[1]dK[1]+c_2 x+c_1\right ) \]

✓ Sympy. Time used: 14.568 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - asin(sin(x)) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \int e^{- 2 x} \operatorname {asin}{\left (\sin {\left (x \right )} \right )}\, dx\right ) - \int x e^{- 2 x} \operatorname {asin}{\left (\sin {\left (x \right )} \right )}\, dx\right ) e^{2 x} \]