Question

The differential equation (5+6−4) +(6+5+2)=0 is ………
Linear differential equation
Bernoulli’s differential equation
Exact differential equation
Cauchy-Euler differential equation

Answer 1

Answer:

In this section we are going to take a look at differential equations in the form,

y′+p(x)y=q(x)yny′+p(x)y=q(x)yn

where p(x)p(x) and q(x)q(x) are continuous functions on the interval we’re working on and nn is a real number. Differential equations in this form are called Bernoulli Equations.

First notice that if n=0n=0 or n=1n=1 then the equation is linear and we already know how to solve it in these cases. Therefore, in this section we’re going to be looking at solutions for values of nn other than these two.

In order to solve these we’ll first divide the differential equation by ynyn to get,

y−ny′+p(x)y1−n=q(x)y−ny′+p(x)y1−n=q(x)

We are now going to use the substitution v=y1−nv=y1−n to convert this into a differential equation in terms of vv. As we’ll see this will lead to a differential equation that we can solve.

We are going to have to be careful with this however when it comes to dealing with the derivative, y′y′. We need to determine just what y′y′ is in terms of our substitution. This is easier to do than it might at first look to be. All that we need to do is differentiate both sides of our substitution with respect to xx. Remember that both vv and yy are functions of xx and so we’ll need to use the chain rule on the right side. If you remember your Calculus I you’ll recall this is just implicit differentiation. So, taking the derivative gives us,

v′=(1−n)y−ny′v′=(1−n)y−ny′

Now, plugging this as well as our