# 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.