Examples of equation in the following topics:

 There are four kinematic equations that describe the motion of objects without consideration of its causes.
 Notice that the four kinematic equations involve five kinematic variables: $d$, $v$, $v_0$, $a$, and $t$.
 Each of these equations contains only four of the five variables and has a different one missing.
 Step two  Find an equation or set of equations that can help you solve the problem.
 Choose which kinematics equation to use in problems in which the initial starting position is equal to zero

 We can prove this simply by integrating the fourth equation over $d^3 {\bf p}$.
 The first two terms yield the lefthand side of the equation above.
 Let's define ${\bf V}=\langle {\bf v} \rangle$ and write out the equation above by components,
 This is the continuity equation.
 Because the fourth equation is consistent with the Lioville equation (seventh equation) and more generally with the Boltzmann equation (sixth equation) and $J^\mu_{;\mu}=0$ if particles are conserved, the Lioville and Boltzmann equations cannot hold if $\nabla_{\bf p} \cdot F \neq 0$ and particles are conserved.

 Identify the problem and solve the appropriate equation or equations for the quantity to be determined.
 Solve the appropriate equation or equations for the quantity to be determined (the unknown).
 We cannot use any equation that incorporates t to find ω, because the equation would have at least two unknown values.
 The equation $\omega 2=\omega 02+2$ will work, because we know the values for all variables except ω.
 Taking the square root of this equation and entering the known values gives

 The wavefunction evolves forward in time according to the timedependent Schrodinger equation
 If the Hamiltonian is independent of time we can solve this equation by
 This realization allows us to write the equation that the wavefunction of an atom must satisfy
 For most atomic states, these effects can be treated at perturbations.We can simplify these equations by using
 This gives the following dimensionless equation

 The Ideal Gas Law is the equation of state of a hypothetical ideal gas.
 Variations of the ideal gas equation may help solving the problem easily.
 Substitute the known values into the equation.
 Choose a relevant gas law equation that will allow you to calculate the unknown variable: We can use the general gas equation to solve this problem: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$.
 Substitute the known values into the equation.

 A next step is to use what is known to find the appropriate equation to find what is unknown.
 While it is easiest to find an equation that leaves only one unknown, sometimes this is not possible.
 In these situations, you can solve multiple equations to find the right answer.
 Remember that equations represent physical principles and relationships, so use the equations and drawings in tandem.
 You may then substitute the knowns into the appropriate equations and find a numerical solution.

 We can write $\displaystyle P' = (\partial P/\partial \rho)_s \rho'$ and rewrite the continuity equation to get
 This is a wave equation with a sound speed of $c_s^2 = (\partial P/\partial \rho)_s$.
 Let's take a solution to this equation for the pressure,

 This simply adds another term to the above version of the Bernoulli equation and results in
 Bernoulli's equation can be applied when syphoning fluid between two reservoirs .
 The Bernoulli equation can be adapted to flows that are both unsteady and compressible.
 However, the assumption of inviscid flow remains in both the unsteady and compressible versions of the equation.
 Adapt Bernoulli's equation for flows that are either unsteady or compressible

 So the vector differential equation governing this simple harmonic motion is:
 In other words this one vector equation is equivalent to three completely separate scalar equations (using $\omega _0 ^2 = k/m$ )
 The equations are uncoupled in the sense that each unknown ( $x,y,z$ ) occurs in only one equation; thus we can solve for $x$ ignoring $y$ and $z$ .
 Plugging these into the equations for $x$ and $y$ gives the two amplitude equations
 We can use the first equation to compute $x_0$ in terms of $y_0$ and then plug this into the second equation to get

 Laplace's equation is important in its own right as the cornerstone of potential theory, but the wave equation also involves the Laplacian derivative, so the ideas discussed in this section will be used to build solutions of the wave equation in spherical and cylindrical coordinates too.
 Before we treat the wave equation, let's look at the simpler problem of Laplace's equation:
 He went on to make profound advances in differential equations and celestial mechanics.
 This equation can be integrated to give: $\phi(x)=ax+b$.
 So in 1D any linear function (or a constant) satisfies Laplace's equation.