Calculus Volume 3

# Key Terms

boundary conditions
the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times
boundary-value problem
a differential equation with associated boundary conditions
characteristic equation
the equation $aλ2+bλ+c=0aλ2+bλ+c=0$ for the differential equation $ay″+by′+cy=0ay″+by′+cy=0$
complementary equation
for the nonhomogeneous linear differential equation
$a2(x)y″+a1(x)y′+a0(x)y=r(x),a2(x)y″+a1(x)y′+a0(x)y=r(x),$

the associated homogeneous equation, called the complementary equation, is
$a2(x)y″+a1(x)y′+a0(x)y=0a2(x)y″+a1(x)y′+a0(x)y=0$
homogeneous linear equation
a second-order differential equation that can be written in the form $a2(x)y″+a1(x)y′+a0(x)y=r(x),a2(x)y″+a1(x)y′+a0(x)y=r(x),$ but $r(x)=0r(x)=0$ for every value of $xx$
linearly dependent
a set of functions $f1(x),f2(x),…,fn(x)f1(x),f2(x),…,fn(x)$ for which there are constants $c1,c2,…cn,c1,c2,…cn,$ not all zero, such that $c1f1(x)+c2f2(x)+⋯+cnfn(x)=0c1f1(x)+c2f2(x)+⋯+cnfn(x)=0$ for all x in the interval of interest
linearly independent
a set of functions $f1(x),f2(x),…,fn(x)f1(x),f2(x),…,fn(x)$ for which there are no constants $c1,c2,…cn,c1,c2,…cn,$ such that $c1f1(x)+c2f2(x)+⋯+cnfn(x)=0c1f1(x)+c2f2(x)+⋯+cnfn(x)=0$ for all x in the interval of interest
method of undetermined coefficients
a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess
method of variation of parameters
a method that involves looking for particular solutions in the form $yp(x)=u(x)y1(x)+v(x)y2(x),yp(x)=u(x)y1(x)+v(x)y2(x),$ where $y1y1$ and $y2y2$ are linearly independent solutions to the complementary equations, and then solving a system of equations to find $u(x)u(x)$ and $v(x)v(x)$
nonhomogeneous linear equation
a second-order differential equation that can be written in the form $a2(x)y″+a1(x)y′+a0(x)y=r(x),a2(x)y″+a1(x)y′+a0(x)y=r(x),$ but $r(x)≠0r(x)≠0$ for some value of $xx$
particular solution
a solution $yp(x)yp(x)$ of a differential equation that contains no arbitrary constants
RLC series circuit
a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit
simple harmonic motion
motion described by the equation $x(t)=c1cos(ωt)+c2sin(ωt),x(t)=c1cos(ωt)+c2sin(ωt),$ as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely
a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution
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