# The order of differential equation is always - Order of a differential equation .

### 4.1 Basics of Differential Equations

This is an example of a general solution to a differential equation.

### Differential Equations

An example of this is given by a mass on a spring.

### Differential Equations

Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution.

This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.

I suppose the first equation has both order and degree defined.

The only difference between these two solutions is the last term, which is a constant.

However, if we already know one solution to the differential equation we can use the method that we used in the last to find a second solution.

### Differential Equations

Description: The same is true in general.

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This relation is called the general solution of differential equations, and it also includes an arbitrary constant.
Reduction of order, the method used in the previous example can be used to find second solutions to differential equations.
Linear differential equations frequently appear as to nonlinear equations.

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In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
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is not a property limited only to a second order equation. It, however, does not hold, in general, for solutions of a nonhomogeneous linear equation.) Note: However, while the general solution of y″ + p(t) y′ + q(t) y = 0 will always be in the form of C1 y1 + C2 y2, where y1 and y2 are some solutions of the equation, the converse is not.
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Reduction of order, the method used in the previous example can be used to find second solutions to differential equations. However, this does require that we already have a solution and often finding that first solution is a very difficult task and often in the process of finding the first solution you will also get the second solution without needing to resort to reduction of order.
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Definition 17.1.8 A first order differential equation is separable if it can be written in the form $\dot{y} = f(t) g(y)$. $\square$ As in the examples, we can attempt to solve a separable equation by converting to the form $$\int {1\over g(y)}\,dy=\int f(t)\,dt.$$ This technique is called separation of variables .