Key Equations

Key Equations

Linear second-order differential equation	$a_2\left(x\right)y^{″}+a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=r\left(x\right)$
Second-order equation with constant coefficients	$a{y}^{″}+b{y}^{\prime }+cy=0$

Complementary equation	$a_2\left(x\right)y^{″}+a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=0$
General solution to a nonhomogeneous linear differential equation	$y(x)=c_1{y}_1(x)+c_2{y}_2(x)+y_p(x)$

Equation of simple harmonic motion	$x^{″}+{\omega }^{2}x=0$
Solution for simple harmonic motion	$x(t)=c_1\text{cos}\left(\omega t\right)+c_2\text{sin}\left(\omega t\right)$
Alternative form of solution for SHM	$x(t)=A\text{sin}\left(\omega t+\varphi \right)$
Forced harmonic motion	$m{x}^{″}+b{x}^{\prime }+kx=f(t)$
Charge in a RLC series circuit	$L\frac{d^{2}q}{d{t}^2}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)$