Deferential Equations

Deferential Equations

First order D.E.

1)Separation of variables

Some differential equations can be solved by the method of separation of variables (or "variables separable") . This method is only possible if we can write the differential equation in the form

A(x) dx + B(y) dy = 0,

where A(x) is a function of x only and B(y) is a function of y only.

Once we can write it in the above form, all we do is integrate throughout, to obtain our general solution.

a) The differential equation

can be expressed in the required form:

Here, A(x) = -1/(x ln x) and B(y) = 1/y.
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b) Solve the equation
    2 y dy = ( x² + 1) dx.
Since this equation is already expressed in “separated” form, just integrate:


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c) Solve the equation 

The equation can be rewritten as follows:


Integrating both sides yields


Since the initial condition states that y = 1 at x = 0, the parameter c can be evaluated:


The solution of the equation is therefore