# First Order Non-linear Differential Equation

The dependence of the velocity v of a particle upon time t obeys the differential equation:

dv/dt = -av^2 - bv

where a>0, b>0 are constants. The initial condition is v(0) = V0 where V0 is the velocity at time t=0.

a. What is the order of this equation? Is it linear or non-linear?

b. Find the general solution of the equation in the case where a=0, b=/=0 and use this to find a solution satisfying the initial condition

c. Find the general solution of the equation in the case where b=0, a=/=0 and use this to find a solution satisfying the initial condition.

d. Write 1/(av^2 + bv) = A/v + B/(av+b) and determine the coefficients A and B.

e. Hence or otherwise find a general solution to the equation for v. Show that your solution leads to the following equation for v(t) which satisfies the stated initial condition:

(aV0 + b)/V0 exp (bt) = (av(t) + b)/ v(t)

f. Rearrange this expression to find an expression for v(t).

See attachment for better symbol representation.

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The order of the equation is determined by the highest order of the derivative.

In this case the highest order of the derivative is 1, therefore this is a first order equation.

Let be a solution of the equation

The equation is linear if is a solution as well, where is an arbitrary constant.

The equation is:

(1.1)

When we plug (1.1) into (1.2) we get:

(1.2)

Obviously, ...

#### Solution Summary

The solution shows in detail how to separate the non-linear equation and use partial fraction method to integrate it.

Contrasting Ordinary and Partial Derivatives

Please also explain when we should use ordinary derivatives and when we should use partial derivatives.

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