Abstract
A new transformation technique to find the analytical solution of general second order linear ordinary differential equation
Highlights
P(x)u′′ + Q(x)u′ + R(x)u = 0 (1)where u = h(x) is the solution of Eq 1 and the coefficients P(x), Q(x) and R(x) are the functions of x and P(x) ≠ 0
If we restrict the coefficient functions of Eq 1 as P(x) = x2, Q(x) = xand R(x) = 1 continuous over the interval (0, ∞) Eq 1 becomes well known Cauchy-Euler equation which can be solved by transformation method by using the transformation u = xm, see Zill (2016)
All the above literature and discussion shows the importance of Eqs. 1 and 2 for their applications but at the same time we can observe that every method to find the general solution of these equations has its own limitations and specific area of implementation based on the nature of coefficient functions, see Al Bastami et al (2010), Batiha (2015), Busawon and Johnson (2005), Johnson et al (2008), Kim (2016), Mohammed and Zeleke (2015), Pala and Ertas (2017), Robin (2007), Sarafian (2011), Saravi (2012), and Wilmer III and Costa (2008)
Summary
All the above literature and discussion shows the importance of Eqs. 1 and 2 for their applications but at the same time we can observe that every method to find the general solution of these equations has its own limitations and specific area of implementation based on the nature of coefficient functions, see Al Bastami et al (2010), Batiha (2015), Busawon and Johnson (2005), Johnson et al (2008), Kim (2016), Mohammed and Zeleke (2015), Pala and Ertas (2017), Robin (2007), Sarafian (2011), Saravi (2012), and Wilmer III and Costa (2008). We have discussed the special case of second order OLDE on the basis of a new transformation to obtain its general solution.
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