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What is a differential equation?
An equation that involves one or more derivatives of an unknown function is called a differential equation. The order of the highest derivative included in a differential equation defines the order of this equation.
Examples
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y ' = 3x ,
the order of the highest derivative is 1 (y ' ) so the order of this differential equation is 1.
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y '' + y' + y = 3x ,
the order of the highest derivative is 2 (y '' ) so the order of this differential equation is 2.
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-2 y ''' + y'' + y 4 = 3x ,
the order of the highest derivative is 3 (y ''' ) so the order of this differential equation is 3.
y = f(x) is a solution of a differential equation if the equation is satisfied upon substitution of y and its derivatives into the differential equation.
Example:
Verify that y = C*e 4x + e 3x, where c is a constant, is a solution to the differential equation
y ' - 4y = -e 3x
y ' is given by
y ' = 4C*e 4x + 3e 3x
We now substitute y ' and y into the left side of the equation and simplify
y ' - 4y = 4C*e 4x + 3e 3x - 4 (C*e 4x + e 3x)
= 4C*e 4x + 3e 3x - 4C*e 4x - 4e 3x
= 4C*e 4x - 4C*e 4x + e 3x (3 - 4)
= - e 3x
Which is equal to the left side of the given equation and therefore y = C*e 4x + e 3x is a solution to the differential equation y ' - 4y = -e 3x.
Most of the work on differential equations consists in solving these equations. For example to solve the following differential example
y ' = 2x
Let us integrate both sides of the given equation as follows
ò y ' dx = ò 2x dx
which gives
y + C1 = x 2 + C2
where C1 and C2 are constants of integration. The solution y of the above equation is given by: y = x 2 + C, where C = C2 - C1.
More references on
Differential Equations
Differential Equations - Runge Kutta Method
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