81.14.10 problem 18-20

Internal problem ID [21695]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 18. Algebra of differential operators. Page 435
Problem number : 18-20
Date solved : Thursday, October 02, 2025 at 08:00:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 a y^{\prime }+\left (a^{2}+b^{2}\right ) y&=f \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (x_{0} \right )&=y_{0} \\ y^{\prime }\left (x_{0} \right )&=y_{1} \\ \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 119

ode:=diff(diff(y(x),x),x)-2*a*diff(y(x),x)+(a^2+b^2)*y(x) = f(x); 
ic:=[y(x__0) = y__0, D(y)(x__0) = y__1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{a x} \left (\int _{x_{0}}^{x}{\mathrm e}^{-a \textit {\_z1}} \cos \left (b \textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \sin \left (b x \right )-\int _{x_{0}}^{x}{\mathrm e}^{-a \textit {\_z1}} \sin \left (b \textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \cos \left (b x \right )+\left (-\left (\left (a y_{0} -y_{1} \right ) \cos \left (b x_{0} \right )-\sin \left (b x_{0} \right ) b y_{0} \right ) \sin \left (b x \right )+\left (\left (a y_{0} -y_{1} \right ) \sin \left (b x_{0} \right )+\cos \left (b x_{0} \right ) b y_{0} \right ) \cos \left (b x \right )\right ) {\mathrm e}^{-a x_{0}}\right )}{b} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 301

ode=D[y[x],{x,2}]-2*a*D[y[x],x]+(a^2+b^2)*y[x]==f[x]; 
ic={y[x0]==y0,Derivative[1][y][x0] ==y1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{a (x-\text {x0})-i b (x+\text {x0})} \left (2 b e^{\text {x0} (a+i b)} \int _1^x\frac {i e^{-((a-i b) K[1])} f(K[1])}{2 b}dK[1]+2 b e^{a \text {x0}+i b (2 x+\text {x0})} \int _1^x-\frac {i e^{-((a+i b) K[2])} f(K[2])}{2 b}dK[2]-2 b e^{a \text {x0}+i b (2 x+\text {x0})} \int _1^{\text {x0}}-\frac {i e^{-((a+i b) K[2])} f(K[2])}{2 b}dK[2]-2 b e^{\text {x0} (a+i b)} \int _1^{\text {x0}}\frac {i e^{-((a-i b) K[1])} f(K[1])}{2 b}dK[1]+i a \text {y0} e^{2 i b x}-i a \text {y0} e^{2 i b \text {x0}}+b \text {y0} e^{2 i b x}-i \text {y1} e^{2 i b x}+b \text {y0} e^{2 i b \text {x0}}+i \text {y1} e^{2 i b \text {x0}}\right )}{2 b} \end{align*}

✓ Sympy. Time used: 1.739 (sec). Leaf size: 233

from sympy import * 
x = symbols("x") 
a = symbols("a") 
x0 = symbols("x0") 
y0 = symbols("y0") 
y1 = symbols("y1") 
y = Function("y") 
f = Function("f") 
ode = Eq(-2*a*Derivative(y(x), x) + (a**2 + b**2)*y(x) - f(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(x0): y0, Subs(Derivative(y(x), x), x, x0): y1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {i a y_{0} e^{- a x_{0}} e^{- i b x_{0}}}{2 b} + \frac {y_{0} e^{- a x_{0}} e^{- i b x_{0}}}{2} - \frac {i y_{1} e^{- a x_{0}} e^{- i b x_{0}}}{2 b} + \frac {i \int f{\left (x_{0} \right )} e^{- a x_{0}} e^{- i b x_{0}}\, dx_{0}}{2 b}\right ) e^{x \left (a + i b\right )} + \left (- \frac {i a y_{0} e^{- a x_{0}} e^{i b x_{0}}}{2 b} + \frac {y_{0} e^{- a x_{0}} e^{i b x_{0}}}{2} + \frac {i y_{1} e^{- a x_{0}} e^{i b x_{0}}}{2 b} - \frac {i \int f{\left (x_{0} \right )} e^{- a x_{0}} e^{i b x_{0}}\, dx_{0}}{2 b}\right ) e^{x \left (a - i b\right )} + \frac {i e^{x \left (a - i b\right )} \int f{\left (x \right )} e^{- a x} e^{i b x}\, dx}{2 b} - \frac {i e^{x \left (a + i b\right )} \int f{\left (x \right )} e^{- a x} e^{- i b x}\, dx}{2 b} \]