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Hello guys I'm struggling to solve this Bernoulli differential equation

we are told to prove that the particular answer YP to the non homogeneous equation is YP=2x



Hello guys I'm struggling to solve this Bernoulli differential equation we are told-example-1

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Answer:

Explanation:

The Bernoulli differential equation is given by:

y' + p(x)y = q(x)y^n

where n is a constant, not equal to 0 or 1. In this case, we have:

y' + (x-3)/x y = (y+x)/x^2 y^2

To solve this equation, we can use the substitution:

u = y^(1-n)

So that:

y = u^(1/(1-n))

y' = (1/(1-n))u^(1/(1-n)-1)u'

Substituting these expressions into the Bernoulli equation and simplifying, we get:

(1/(1-n))u^(1/(1-n)-1)u' + (x-3)/x u^(1/(1-n)) = (u^(1/(1-n))+x)/x^2 (u^(2/(1-n)))

Multiplying both sides by (1-n)u^(1/(1-n)) and simplifying, we get:

(1-n)u^(1/(1-n))u' + (x-3)/x (1-n)u = (u^(1/(1-n))+x)/x^2 (u^(2/(1-n)+1))

The left-hand side is the derivative of (1-n)u^(1/(1-n)) with respect to u, so we can write:

d/dx[(1-n)u^(1/(1-n))] = (u^(1/(1-n))+x)/x^2 (u^(2/(1-n)+1))

Integrating both sides with respect to u, we get:

(1-n)u^(1/(1-n)) = (1/(2/(1-n)+1))u^(2/(1-n)+1) + C

Substituting u = y^(1-n), we get:

(1-n)y = (1/(2/(1-n)+1))y^(2/(1-n)+1) + C

To find the particular solution to the non-homogeneous equation, we can use the method of undetermined coefficients and guess that the particular solution has the form YP = ax + b. Then:

YP' = a

YP^2 = a^2x^2 + 2abx + b^2

Substituting these expressions into the non-homogeneous equation and simplifying, we get:

2a^2x^2 + (2ab - a)x + (b^2 - b) = 0

For this equation to hold for all x, we must have:

2a^2 = 0

2ab - a = 1

b^2 - b = 0

The first equation gives a = 0, and substituting this into the second equation gives b = 1. Therefore, the particular solution is:

YP = 2x

To find the general solution to the non-homogeneous equation, we add the particular solution to the homogeneous solution we found earlier:

y = (1/(1-n))u^(1/(1-n)) + 2x

Substituting u = y^(1-n), we get:

y = (1/(1-n))(y^(1-n))^(1/(1-n)) + 2x

Simplifying:

y = ((1-n)y)^(-1/n) + 2x

Therefore, the general solution to the non-homogeneous equation is:

y = ((1-n)y)^(-1/n) + 2x

Note that this solution is only valid for n not equal to 1 or 0, since those cases correspond to the linear and separable differential equations, respectively.

User Tomasz Grobelny
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