Examples of differential equation in the following topics:

 Elementary differential equations and boundary value problems.

 This is an example of a constant coefficient differential equation.
 ($x$ is the zeroth derivative of $x$. ) The most general linear nth order constant coefficient differential equation is
 These constant coefficient differential equations have a very special property: they reduce to polynomials for exponential $x$.
 Without $i$, not even a simple equation such as $\ddot{x} +x =0$ has a solution.
 With $i$ every algebraic equation can be solved.

 The equation describing the evolution of the radiation field is still rather innocuous looking
 The evolution of the intensity of a particular ray depends not only the intensity of the ray and the local properties of the material but also on the intensity of all other rays passing through the same point—we have an integrodifferential equation.
 Even if one neglects scattering, one often has to solve an integrodifferential equation.

 In this equation, dl represents the differential of length of wire in the curved wire, and μ0 is the permeability of free space.
 In this equation, the r vector can be written as r̂ (the unit vector in direction of r), if the r3 term in the denominator is reduced to r2 (this is simply reducing like terms in a fraction).
 Integrating the previous differential equation, we find:
 Express the relationship between the strength of a magnetic field and a current running through a wire in a form of equation

 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.
 When solving boundary value problems for differential equations like Laplace's equation, it is extremely handy if the boundary on which you want to specify the boundary con ditions can be represented by holding one of the coordinates constant.
 The disadvantage to using coordinate systems other than Cartesian is that the differential operators are more complicated.
 This equation can be integrated to give: $\phi(x)=ax+b$.

 Kirchhoff's circuit laws are two equations that address the conservation of energy and charge in the context of electrical circuits.
 Kirchhoff's circuit laws are two equations first published by Gustav Kirchhoff in 1845.
 Although Kirchhoff's Laws can be derived from the equations of James Clerk Maxwell, Maxwell did not publish his set of differential equations (which form the foundation of classical electrodynamics, optics, and electric circuits) until 1861 and 1862.

 Maxwell's equations help form the foundation of classical electrodynamics, optics, and electric circuits.
 Maxwell's equations are a set of four partial differential equations that, along with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits.
 Maxwell's equations can be divided into two major subsets.
 The differential form of Gauss's law for magnetic for magnetism is
 Both macroscopic and microscopic differential equations are the same, relating electric field (E) to the timepartial derivative of magnetic field (B):

 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

 ., we consider what happens when there is no vector that satisfies the equations exactly.
 You can think of this as $n$ equations with 1 unknown:
 Differentiating this equation with respect to $x$ and setting the result equal to zero gives:
 These are called the normal equations.
 differentiating $(A\mathbf{x}, A\mathbf{x})$ with respect to $2 A^TA x$ yields $2 A^TA x$ , one factor coming from each factor of $\mathbf{x}$ .

 There are a number of different techniques for solving the 1D wave equation:
 where the prime denotes differentiation with respect to the argument ( $x+ct$ for instance).
 As a result, any differentiable function evaluated at $x \pm ct$ is a solution of the 1D wave equation.