# Linear Differential Equations of the 1st to 4th Order

For linear differential equations with constant coefficients, the *analytic solution* is generated and displayed graphically.

The independent variable here is x, the dependent variable is y, i.e. y = y(x).

Example of an inhomogeneous differential equation of 2nd order:

y'' + 2y' + 5y = sin(3x)

For the particular solution of the inhomogeneous equation, the usual approach technique is used, which is based on the type on the right side.

Allowed right-hand pages are:
a·cos(b·x), a·sin(b·x), a·exp(b·x) und a·x^{c} with a,b ∈ **ℝ** and c ∈ **ℕ₀**.

For the *initial value problem*, n initial conditions for an initial x-value, e.g. x=0, must be created for an nth-order equation:

y(0)=r_{0}, y'(0)=r_{1}, ... y^{(n-1)}(0)=r_{n-1} mit r_{i} ∈ **ℝ**

With this, the free coefficients C_{i} of the general solution of the homogeneous equation are determined, taking into account the particular solution.

In a *boundary value problem*, on the other hand, n specifications for the solution y(x) and/or its derivatives are made at the edges of the area to be examined.

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