23.3.101 problem 103

Internal problem ID [5815]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 103
Date solved : Tuesday, September 30, 2025 at 02:03:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} b y+a y^{\prime }+y^{\prime \prime }&=f \left (x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 110

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

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

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 152

ode=b*y[x] + a*D[y[x],x] + D[y[x],{x,2}] == f[x]; 
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 (\int _1^x-\frac {e^{\frac {1}{2} \left (a+\sqrt {a^2-4 b}\right ) K[1]} f(K[1])}{\sqrt {a^2-4 b}}dK[1]+e^{x \sqrt {a^2-4 b}} \int _1^x\frac {e^{\frac {1}{2} \left (a-\sqrt {a^2-4 b}\right ) K[2]} f(K[2])}{\sqrt {a^2-4 b}}dK[2]+c_2 e^{x \sqrt {a^2-4 b}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 2.233 (sec). Leaf size: 148

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) - f(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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