Ordinary Differential Equations/Bernoulli

An equation of the form

${dy \over dx}+ f(x)y = g(x)y^n$

can be made linear by the substitution $z=y^{1-n}$

Its derivative is

${dz \over dx}=(1-n)y^{-n}{dy \over dx}$

So that multiplying it by $y^{-n}$

The equation can be turned into

${dy \over dx}y^{-n} + f(x)y^{1-n} = g(x)$

${dz \over dx} + (1-n)f(x)z = (1-n)g(x)$

Which is linear.

The Jacobi equation

$(a_1+b_1x+c_1y)(xdy-ydx)-(a_2+b_2x+c_2y)dy+(a_3+b_3x+c_3y)dx=0$

can be turned into the Bernoulli equation with the appropriate substitutions.

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