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Question

To reduce the differential equation dydx+P(x)·y=Q(x)·yn to the linear form, the substitution is


A

v=1yn

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B

v=1yn-1

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C

v=yn

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D

v=yn-1

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Solution

The correct option is B

v=1yn-1


Explanation for the correct option:

Find the substitution to reduce the given differential equation in the linear differential equation.

In the question, an equation dydx+P(x)·y=Q(x)·yn is given. This is also known as Bernoulli's equation.

Divide both sides of the given differential equation by yn.

d(y-n+1)dx+P(x)·y-n+1=Q(x)(-n+1)y-ndydx+P(x)·y-n+1=Q(x)...1

Assume that, v=y-n+1.

Differentiate both sides with respect to x.

dvdx=(-n+1)y-ndydx

So, equation 1 becomes dvdx+P(x)·v=Q(x) which is a linear differential equation.

Therefore, to reduce the differential equation dydx+P(x)·y=Q(x)·yn to the linear form, the substitution is v=1yn-1.

Hence, option B is the correct option.


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