Mathematics Differential Equations Second-Order Differential Equation | linear-nonlinear | Homo-Non Homogeneous

Second-Order Differential Equation | linear-nonlinear | Homo-Non Homogeneous

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In previous discussion we have talked about the first order differential equations, see here ». In this article we will discuss about the linear Second-Order Differential Equation. The order of such equation, n=2. Consider a second-order differential equation of unknown function $y(x)$, takes the form:

$$G(x, y, y^{′}, y^{″}) = 0$$

The second-order differential equation can also be linear, because $G$ is linear, which depends on the dependent variable $y$ and its derivatives $y^{′},$ and $y^{″}$. The general form of such equation can be written as,

$$A(x)y^{″}+B(x)y^{′}+C(x)y = F(x)\quad\text{…eqn(1)}.$$

The Coefficient functions, $A(x),$ $B(x),$ $C(x)$ and $F(x)$ are continuous on some open interval $I$, on which we wish to solve this differential equation. But, we don’t require these coefficients to be linear functions of $x$. Thus, such equation,

$$e^{x}y^{″}+(cos x)y^{′}+(1+\sqrt{x})y = tan^{-1}\quad\text{…eqn(2)}.$$

Second-Order Differential Equation | Linearity and non-linearity


The above equation in eqn(2), is a linear second-order differential equation. Because the highest derivative appears in the equation is second derivative. -and the equation is a Linear differential equation, because, the dependent variable $y,$ and its derivatives $y^{′},$ and $y^{″}$ appear in the equation raised to its first power$^{1}$. The eqn(1), is linear, also because, no product of derivatives (like – $y.y^{′}$) appears.

NON-LINEARITY


In case of non-linearity, Consider the second-order differential equation,

$$y^{″}+3(y^{′})^{2}+4y^{3} = 0$$

which is non-linear. Because, the power of dependent variable $y$ and its derivative appear. Again, the differential equation, $y^{″} – yy^{′} = 0,$ is also non-linear, because, the product of $y,$ and its derivative appears.

Homogeneous – Non Homogeneous


The second-order differential equation is either Homogeneous or Non Homogeneous. It depends on the Non-Homogeneous term, $F(x).$ If the function $F(x)$ on the right-hand-side of the eqn(1), vanishes identically on open interval $I,$ then we call the equation is a Homogeneous linear equation. Otherwise, the equation is nonhomogeneous. For example, the second-order differential equation,

$$x^{2}y^{″}+2xy^{′}+3y = cos x,$$

is a nonhomogeneous equation. Its associated homogeneous equation is,

$$x^{2}y^{″}+2xy^{′}+3y = 0.$$

Therefore, the general form of homogeneous linear equation is,

$$A(x)y^{″}+B(x)y^{′}+C(x)y = 0\quad\text{…eqn(3)}.$$

Solution of second-order differential equation


Solution of homogeneous equation very simple. But, the solution of nonhomogeneous second-order differential equation consists of two parts. Those are: $y_{c},$ which is the solution of complementary homogeneous equation, and $y_{p},$ is the particular solution. The particular solution consists of a method called undetermined coefficient. -which is a guessing technique, we need to guess the form of particular equation, and then find the solution for $x.$

In the coming lessons, we will discuss about both of them in greater details.

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