ABSTRACT
Bernoulli equations discussed in the previous chapter are the first serious non-linear odes taken up. However, the generalised parabolic or hyperbolic transformation enabled us to recast the Bernoulli equation to linear odes. The non-linearity in Bernoulli’s equation owed its origin to the terms xαyβ as the derivative-free terms. There are also many non-linear first order differential equations whose non-linearity is due to the occurence of higher powers of https://www.w3.org/1998/Math/MathML"> d y d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003205982/63298289-b4dc-4f72-9a4c-f5000312b2eb/content/math3_130_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (p is the conventional abridged notation for https://www.w3.org/1998/Math/MathML"> d y d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003205982/63298289-b4dc-4f72-9a4c-f5000312b2eb/content/math3_130_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ). The general form of the first order n-th degree ode reads https://www.w3.org/1998/Math/MathML"> p n + P 1 p n − 1 + P 2 p n − 2 + ... + P n − 1 p + P n = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003205982/63298289-b4dc-4f72-9a4c-f5000312b2eb/content/math3_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where Pk ’s are functions of x and y.
