55.28.1 problem 11

Internal problem ID [13784]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 11
Date solved : Thursday, October 02, 2025 at 08:06:41 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} b y+a y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)+a*diff(y(x),x)+b*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 \,{\mathrm e}^{x \sqrt {a^{2}-4 b}}+c_2 \right ) {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 b}+a\right )} \left (c_2 e^{x \sqrt {a^2-4 b}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.146 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} e^{\frac {x \left (- a + \sqrt {a^{2} - 4 b}\right )}{2}} + C_{2} e^{- \frac {x \left (a + \sqrt {a^{2} - 4 b}\right )}{2}} \]