Abernoulli differential equation is one of the form

dy/dx+p(x)y=q(x)y^n.

observe that, if n=0 or 1, the bernoulli equation is linear. for other values of n, the substitution u=y1−n transforms the bernoulli equation into the linear equation

du/dx+(1−n)p(x)u=(1−n)q(x).

use an appropriate substitution to solve the equation

y′−5/x*y=y^5/x^9,

and find the solution that satisfies y(1)=1.

 The solution to the differential equation y′−5/x*y=y^5/x^9, by using the appropriate substitution is . The general solution that

satisfies y(1) = 1 is

The given equation is well expressed as:

We are to apply the Bernoulli equation approach to solve the given differential equation.

So, the first step is to divide the differential equation by , by doing so, we have

Now, let make:

∴

From the equation   ;

We can now have the equation to be differentiated as:

Hence, we can infer that the linear equation in terms of u is:

Now, taking the integration factor:

We need to multiply the integration factor with the equation

, so we have:

Taking the integral of both sides;

Recall that:  , if we replace the value of u with in the above equation, we will have:

Using the given initial condition y(1) = 1  to the equation above to determine the value of c, we have:

∴

Now, to solve for y(x); we have:

Therefore, the general solution that  satisfies y(1) = 1 is

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Divide through both sides by :

Now let , so that . Then

Multiply both sides by :

Integrate both sides to get

Solve for :

Given that , we have

so the particular solution is

which we can rewrite as

step-by-step explanation:

the correct answer is a.) bc< mn 5 stars if this you : )

step-by-step explanation: