Variation of Parameter Method for Solving Homogeneous Second Order Linear Ordinary Differential Equations

A differential equation is an equation that relates an unknown function and one or more of its derivatives of with respect to one or more independent variables [1]. If the unknown function depends only on a single independent variable, such a differential equation is ordinary differential equation. The order of an ordinary differential equation is the order of the highest derivative that appears in the equation [2]. A differential equation with an unknown function is said to be linear if it is linear in that unknown function and its derivatives. A differential equation of the form ay(2)+by(1)+cy=f(x) is said to be second order linear ordinary differential equation with constant coefficients if a,b and c are constants and a is non zero scalar, where f(x) is a function of x. The associated homogeneous equation of second order linear ordinary differential equation with constant coefficients ay(2)+by(1)+cy=f(x) is ay(2)+by(1)+cy=0. First we solve equation ay(2)+by(1)+cy=0 to solve second order linear ordinary differential equation with constant coefficients ay(2)+by(1)+cy=f(x). The polynomial ap2+bp+c is called the characteristic polynomial of ay(2)+by(1)+cy=0 [3]. The function = exp y mdx ∫ is solution of ay(2)+by(1)+cy=0 if and only if m is root of the characteristic polynomial ap2+bp+c [3]. Thus, the solution of equation ay(2)+by(1)+cy=0 is given by = exp y mdx ∫ whenever it is second order linear ordinary differential equation with constant coefficients, where m is a number such that am2+bm+c=0. We solve the characteristic equation am2+bm+c=0 of ay(2)+by(1)+cy=0 to solve ay(2)+by(1)+cy=0. Clearly the characteristic equation am(2)+bm+c=0 of ay(2)+by(1)+cy=0 is quadratic equation in m. Thus, we consider three cases for solution of the characteristic equation am2+bm+c=0 of ay(2)+by(1)+cy=0. Most authors of differential books used this technique to derive solution method for solving second order linear ordinary differential equation with constant coefficients. We apply Moivre’s theorem to find general solution of ay(2)+by(1)+cy=0 when b2-4ac<0 [4]. In this manuscript, we introduce variation of parameter method to find general solution of homogeneous second order linear ordinary differential equation with constant coefficients. Furthermore, we do not use Moivre’s theorem to find general solution of homogeneous second order linear ordinary differential equation with constant coefficients. If we replace a scalar m by a function u(x) and assume that = exp ( ) y u x dx ∫ is solution of homogeneous second order linear ordinary differential equation with constant coefficients, then = exp ( ) y u x dx ∫ is variation of parameter method for solving second order linear ordinary differential equation with constant coefficients, where u(x) is a function of x to be determined. Finally, we determine u(x) by assuming = exp ( ) y u x dx ∫ as solution of homogeneous second order linear ordinary differential equation with constant coefficients.


Introduction
A differential equation is an equation that relates an unknown function and one or more of its derivatives of with respect to one or more independent variables [1]. If the unknown function depends only on a single independent variable, such a differential equation is ordinary differential equation. The order of an ordinary differential equation is the order of the highest derivative that appears in the equation [2]. A differential equation with an unknown function is said to be linear if it is linear in that unknown function and its derivatives. A differential equation of the form ay (2) +by (1) +cy=f(x) is said to be second order linear ordinary differential equation with constant coefficients if a,b and c are constants and a is non zero scalar, where f(x) is a function of x. The associated homogeneous equation of second order linear ordinary differential equation with constant coefficients ay (2) +by (1) +cy=f(x) is ay (2) +by (1) +cy=0. First we solve equation ay (2) +by (1) +cy=0 to solve second order linear ordinary differential equation with constant coefficients ay (2) +by (1) +cy=f(x). The polynomial ap 2 +bp+c is called the characteristic polynomial of ay (2) +by (1) +cy=0 [3]. The function = exp y mdx ∫ is solution of ay (2) +by (1) +cy=0 if and only if m is root of the characteristic polynomial ap 2 +bp+c [3]. Thus, the solution of equation ay (2) +by (1) +cy=0 is given by = exp y mdx ∫ whenever it is second order linear ordinary differential equation with constant coefficients, where m is a number such that am 2 +bm+c=0. We solve the characteristic equation am 2 +bm+c=0 of ay (2) +by (1) +cy=0 to solve ay (2) +by (1) +cy=0. Clearly the characteristic equation am (2) +bm+c=0 of ay (2) +by (1) +cy=0 is quadratic equation in m. Thus, we consider three cases for solution of the characteristic equation am 2 +bm+c=0 of ay (2) +by (1) +cy=0. Most authors of differential books used this technique to derive solution method for solving second order linear ordinary differential equation with constant coefficients. We apply Moivre's theorem to find general solution of ay (2) +by (1) +cy=0 when b 2 -4ac<0 [4]. In this manuscript, we introduce variation of parameter method to find general solution of homogeneous second order linear ordinary differential equation with constant coefficients. Furthermore, we do not use Moivre's theorem to find general solution of homogeneous second order linear ordinary differential equation with constant coefficients. If we replace a scalar m by a function u(x) and assume that = exp ( ) y u x dx ∫ is solution of homogeneous second order linear ordinary differential equation with constant coefficients, then = exp ( ) y u x dx ∫ is variation of parameter method for solving second order linear ordinary differential equation with constant coefficients, where u(x) is a function of x to be determined. Finally, we determine u(x) by assuming = exp ( ) y u x dx ∫ as solution of homogeneous second order linear ordinary differential equation with constant coefficients.

Motivation Research Questions
is solution of homogeneous second order linear ordinary differential equation with constant coefficients.

Linear first order differential equations
The linear first order ordinary differential equation with unknown dependent variable y and independent variable x is defined by

Linear second order differential equations
The second order linear ordinary differential equation with unknown dependent variable y and independent variable x is defined by (1) (2) 0 1 2 ( ) ( ) ( ) = ( ). a x y a x y a x y g x + + (3)

Fundamental set of solutions:
A set of functions y 1 (x),y 2 (x),…,y n (x) is said to be linearly dependent on an interval I if there exist constants c 1 ,c 2 ,…c n not all zero, such that c 1 y 1 (x)+c 2 y 2 (x)+…+c n y n (x)=0 for every x in the interval. If the set of functions is not linearly dependent on the interval, it is said to be linearly independent. Any set y 1 (x),y 2 (x),…,y n (x) of n linearly independent solutions of the homogeneous linear nth-order differential equation on an interval I is said to be a fundamental set of solutions on the interval [3].
General solution of linear second order differential equations: The associated homogeneous differential equation of nonhomogeneous linear nth order differential equation (1) ( 1) n n n n a x y a x y a x y a x y Dennis − − + + + +  Theorem 1: Let y 1 ,y 2 ,…,y n be linearly independent solutions of the homogeneous linear nth order differential equation Where y 1 and y 2 are linearly independent solutions of the associated homogeneous equation of the equation in 3 and y p is particular solution of the equation in 3. Here c i ,(i=1,2) are arbitrary constants [5].

Construction of a second solution from a known solution: Let
Then we assume that y 2 (x)=u(x)y 1 (x) as another solution of the equation in equation 6 to construct the second solution of the equation in equation 6.

It
follows that is solution of the equation in equation 6, we have (1) (2) 2 2 2 = 0 cy by ay + + From equation in equation 8 it follows that Using equation in equation 7, the equation in equation 9 is reduced to The equation in equation 12 is linear first order ordinary differential equation. Thus, using the formula in equation 2, we get Where y 1 and y 2 are linearly independent solutions of the associated homogeneous equation of the equation in 3 and y p is particular solution of the equation in 3. Here c i ,(i=1,2) are arbitrary constants [5].

General Solution of Homogeneous Second Order Linear Ordinary Differential Equation with Constant Coefficients
Theorem 2: The general solution of second order linear ordinary differential equation ay (2) +by (1) +cy=0 with constant coefficients a,b and c is ∫ as solution of homogeneous second order linear ordinary differential equations with constant coefficients, where m is a scalar to be determined. In this manuscript, We introduce variation of parameter method to prove this theorem. First we assume that = exp ( ) y u x dx ∫ as solution of homogeneous second order linear ordinary differential equations with constant coefficients, where u(x) is a function to be determined. Then we transform a homogeneous second order linear ordinary differential equation with constant coefficients to separable first order ordinary differential equation by this assumption. Finally, we solve an equivalent separable first order ordinary differential equation to prove this theorem.
Suppose that = exp ( ) The equation in 19 is separable first order ordinary differential equation because it is equivalent to  Then the equation in 20 is equivalent to This implies that Thus, Since u=v+α, we have . This implies that . Thus, It follows from equations in 23 and 24 that It follows from equations in 26 and 27 that

Results and Discussion
Most authors of differential equations used Moivre's theorem to find general solution of ay (2) +by (1) +cy=0 when b 2 -4ac<0 [4]. In this manuscript, we introduced variation of parameter method to find general solution of homogeneous second order linear ordinary differential equation with constant coefficients. Furthermore, we found general solution of homogeneous second order linear ordinary differential equation with constant coefficients without applying Moivre's theorem .

Conclusion
In this manuscript, we determined undetermined functions u(x) by assuming = exp ( ) y u x dx ∫ as solution of homogeneous second order linear ordinary differential equation with constant coefficients. Moreover, we proved in theorem 2 by applying variation of parameter method for solving homogeneous second order linear ordinary differential equation with constant coefficients.