Higher Order Differential Equations

Introduction

Thus far we have looked at first-order differential equations. A higher order differential equation will have second, third and up to n derivatives:

\( \cfrac{d^3y}{dx^3} - x\cfrac{d^2y}{dx^2} + 2 = \sin {x} \)

The general form of a higher-order DE is:

\( a_n(x)\cfrac{d^ny}{dx^n} + a_{n-1}(x)\cfrac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x)\cfrac{dy}{dx} + a_0(x)y = g(x)\)

Notice that the coefficients infront of the derivatives are functions of \(x\). For an intial value problem, the DE is subject to:

\( y(x_0) = y_0, \cfrac{dy}{dx}(x_0) = y_1, \dots \cfrac{d^{n-1}y}{dx^{n-1}} (x_0) = y_{n-1}\)

When \(g(x) = 0\), the equation is homogeneous, otherwise the equation is non-homogeneous.

Classify the DE \( \ddot{y} + 2x\dot{y} - e^{x^2} = 0 \).


Superposition Principle

the Superposition Principle states that if \(y_1, y_2, \dots y_n\) are solutions to a homogeneous differential equation on interval \(I\), then so is any linear combination of those solutions:

\( y = \sum {c_iy_i} \)

Show that a linear combination of solutions \(y = e^{-10x}\) and \(y = e^{5x}\) are also solutions to the DE \(y^{''} + 5y^{'} - 50 y = 0 \).


Try these questions: