81.9.6 problem 13-14

Internal problem ID [21595]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 13. The Wronskian and linear independence. Page 283.
Problem number : 13-14
Date solved : Thursday, October 02, 2025 at 07:58:49 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 110

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

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

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 152

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

\begin{align*} y(x)&\to e^{-\frac {1}{2} x \left (\sqrt {b^2-4 c}+b\right )} \left (\int _1^x-\frac {e^{\frac {1}{2} \left (b+\sqrt {b^2-4 c}\right ) K[1]} f(K[1])}{\sqrt {b^2-4 c}}dK[1]+e^{x \sqrt {b^2-4 c}} \int _1^x\frac {e^{\frac {1}{2} \left (b-\sqrt {b^2-4 c}\right ) K[2]} f(K[2])}{\sqrt {b^2-4 c}}dK[2]+c_2 e^{x \sqrt {b^2-4 c}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 2.302 (sec). Leaf size: 148

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

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