# Solving Differential Equations with Substitution and Bernoulli

56. Suppose that n does not equal to zero and n does not equal to one. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv/dx + (1-n)P(x)v(x) = (1-n)Q(x).

63. The equation dy/dx = A(x)y2 + B(x)y + C(x) is called a Riccati equation. Suppose that one particular solution y1(x) of this equation is known.

Show that the substitution y = y1 + (1/v) transforms the Riccati equation into the linear equation dv/dx + (B +2Ay1)v = -A

66. An equation of the form y = xy' + g(y') is called a Clairaut equation. Show that the one parameter family of straight lines described by y(x) = Cx + g(C) is a general solution of y = xy' + g(y').

keywords: ODE, ODEs

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56. Suppose that n does not equal to zero and n does not equal to one. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv/dx + (1-n)P(x)v(x) = (1-n)Q(x).

Proof. Let v = y1- n. Then by the chain law, we can have

So,

..........................(1)

Note that

......................................(2)

We multiply by y- n on both sides of the Bernoulli equation dy/dx + P(x)y = Q(x)yn, we ...

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Differential equations are solved. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.