Bernoulli’s Differential Equation The general form of this type of equations is: y ′ + P (x)y = Q(x)y n (1) with n any real number. The equation is easily solvable for n = 0 or n = 1. For other values of n this equation can be solved using the following substitution: u = y 1-n (2) EXAMPLE 1 . Solve the following Bernoulli’s equation: xy ′ + y = x 2 y 2 The equation can be re-written in form (1) simply dividing by x: y ′ + 1 x y = xy 2 The substitution to be used in this case is u = y 1-2 =1/y, or y =1/u. Given that, y ′ = - 1 u 2 u ′ the initial equation becomes, - 1 u 2 u ′ + 1 x 1 u = x 1 u 2 ⇓ u ′ - 1 x u = -x We have, thus, obtained a first order linear differential equation with P (x)= -1/x and Q(x)= -x. The general solution of this equation is: u = -x 2 + cx To conclude, given that y =1/u, we have the following general solution for the given Bernoulli’s equation: y = 1 -x 2 + cx 1 James Foadi - Oxford 2011