Calculus Volume 3

# Key Equations

Calculus Volume 3Key Equations
• Linear second-order differential equation
$a2(x)y″+a1(x)y′+a0(x)y=r(x)a2(x)y″+a1(x)y′+a0(x)y=r(x)$
• Second-order equation with constant coefficients
$ay″+by′+cy=0ay″+by′+cy=0$
• Complementary equation
$a2(x)y″+a1(x)y′+a0(x)y=0a2(x)y″+a1(x)y′+a0(x)y=0$
• General solution to a nonhomogeneous linear differential equation
$y(x)=c1y1(x)+c2y2(x)+yp(x)y(x)=c1y1(x)+c2y2(x)+yp(x)$
• Equation of simple harmonic motion
$x″+ω2x=0x″+ω2x=0$
• Solution for simple harmonic motion
$x(t)=c1cos(ωt)+c2sin(ωt)x(t)=c1cos(ωt)+c2sin(ωt)$
• Alternative form of solution for SHM
$x(t)=Asin(ωt+ϕ)x(t)=Asin(ωt+ϕ)$
• Forced harmonic motion
$mx″+bx′+kx=f(t)mx″+bx′+kx=f(t)$
• Charge in a RLC series circuit
$Ld2qdt2+Rdqdt+1Cq=E(t)Ld2qdt2+Rdqdt+1Cq=E(t)$