69.15.9 problem 440

Internal problem ID [18247]
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 15.2 Homogeneous differential equations with constant coefficients. Exercises page 121
Problem number : 440
Date solved : Thursday, October 02, 2025 at 03:09:47 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 4 y^{\prime \prime }-8 y^{\prime }+5 y&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

ode:=4*diff(diff(y(x),x),x)-8*diff(y(x),x)+5*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (c_1 \sin \left (\frac {x}{2}\right )+c_2 \cos \left (\frac {x}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 28

ode=4*D[y[x],{x,2}]-8*D[y[x],x]+5*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (c_2 \cos \left (\frac {x}{2}\right )+c_1 \sin \left (\frac {x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.089 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) - 8*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {x}{2} \right )} + C_{2} \cos {\left (\frac {x}{2} \right )}\right ) e^{x} \]