On errors in Euler’s complex exponent and formula for solving ODEs

Abstract

There are some mistakes in Euler’s works. Some of them form the basis of most of the sciences, including differential equations and complex analysis. We discuss them here.

On Solutions of the Nonlinear Heat Equation Using Modified Lie Symmetries and Differentiable Topological Manifolds

“…To use differentiable topological manifolds to solve (54), we first have to make a connection between the elementary mathematical level we are at and equivalent classes, found deep inside differentiable topological manifolds. This author addressed this extensively in [12] and [13]. Here we simply apply what is developed there to (54).…”

An analysis of the potential Korteweg-DeVries equation through regular symmetries and topological manifolds

“…This is a wide spread practise. The third author demonstrated in [4] that this partitioning of solutions tend to lead to faulty conclusions.…”

“…𝑓 3 = 3𝑦4 (𝐶 1 𝑦5 + 6 𝐶 2 𝑦 4 + 12 𝐶 3 𝑦 3 − 48 𝐶 1 𝑦 2 − 144𝐶 2 𝑦 − 144 𝐶 3 )(65536 (𝐶 1 𝑦 5 − 6 𝐶 3 𝑦 3 + 24 𝐶 1 𝑦 2 + 72 𝐶 2 𝑦 + 72 𝐶 3 ) 3 + 1769472 (𝐶 1 𝑦 3 + 3 𝐶 2 𝑦 2 + 3 𝐶 3 𝑦) 2 (𝑔 5 ) + 1769472 (𝐶 1 𝑦 + 2 𝐶 2 )(𝐶 1 𝑦 5 − 6 𝐶 3 𝑦 3 + 24 𝐶 1 𝑦 2 + 72 𝐶 2 𝑦 + 72 𝐶 3 )𝑔 5 + 1179648(𝐶 1 𝑦 3 + 3 𝐶 2 𝑦 2 + 3 𝐶 3 𝑦)(𝐶 1 𝑦 5 − 6 𝐶 3 𝑦 3 + 24 𝐶 1 𝑦 2 + 72 𝐶 2 𝑦 + 72 𝐶 3 )(𝐶 1 𝑦 7 + 3 𝐶 2 𝑦 6 + 3 𝐶 3 𝑦 5 + 45 𝐶 1 2 𝑦 4 + 270 𝐶 1 𝐶 2 𝑦 3 + An analysis of the potential Korteweg-DeVries equation through regular… 61 𝑔 6 = (𝐶 1 𝑦 9 + 6 𝐶 2 𝑦 8 + 12 𝐶 3 𝑦 7 − 48 𝐶 1 𝑦 6 − 144 𝐶 2 𝑦 5 − 144 𝐶 3 𝑦 4 ),…”