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).
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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:
When we plug (1.1) into (1.2) we get:
The solution shows in detail how to separate the non-linear equation and use partial fraction method to integrate it.