is linear. Definition . A non linear ordinary differential equation is simply one that is not linear. If the coefficients of y y y y n ′ ′′ contain the dependent variable y or its derivatives or if powers of y y y y n ′ ′′ , such as ( ) ′ y , appear in the equation, then the differential equation is non linear.
Also, non linear functions of the dependent variable or its derivatives, such as sin y or e y ′ cannot appear in a linear equation. For instance, ( ) dy ax , d y and dy p x y q x ( ) are linear differential equations where as y dy is a non linear differential equation. ( ) ′′+ ′ = x y is a second order linear ODE. ( ) ′′+ ′ = x is a second order linear ODE.
( ) y + ′ = is a first order non linear ODE. ( ) ′ = sin( ) is a first order non linear ODE. ( ) ′′ = sin( ) is a second order linear ODE. Definition .
If g x ( ) = in ( ), then the above equation is said to be homogeneous , otherwise it is called non-homogeneous . Remark If y x i i ( ), = are any two solutions of homogeneous equation x y x y a x y x a x y x ( ) ( ) ' …( ) then a x y x y a x y x a x y x i i i i , i = Suppose u x c y x c y , where c and c are arbitrary constants. Then, it can be easily verified that u x ( ) is also a solution of ( ). Thus, a first order linear differential equation is written as ′+ p x y ( ) .
A first order differential equation that can’t be