Preface
This section presents some basic properties of second and higher order differential equations.
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Glossary
Differential Equations of higher order
Recall from calculus that derivatives of a smooth function y(x) are denoted as \( y'(x) \) (Lagrange notation) or \( {\text d}y /{\text d}x \) (Leibniz notation), or, in case of time variable t, as \( \dot{y} . \) Second derivatives are denoted by \( y''(x) \) or \( {\text d}^2 y /{\text d}x^2 \) or \( \ddot{y} , \) respectively. Following Euler, we will denote by \( \texttt{D} = {\text d}/{\text d}x \) the (unbounded) derivative operator. Usually, the notations x or t stand for the independent variables and will be widely used. Higher order derivatives have similar notation. For instance, \( y^{(4)} (x) \) stands for the fourth derivative of function y(x).
A second order differential equation in the normal form is as follows: \begin{equation} \label{E41.1} \frac{{\text d}^2 y}{{\text d} x^2} = F\left( x,y,\frac{{\text d}y}{{\text d}x} \right) \qquad \mbox{or} \qquad y'' = F\left( x,y,y' \right) , \end{equation} where F(x, y, p) is some given function of three variables. If the function F(x,y,p) is linear in variables y and p (that is, \( F(x,ay_1 + by_2 , p) = a\,F(x,y_1 , p) + b\,F(x, y_2 , p) \) for any constants a, b, and similar for variable p), then \( y'' = F\left( x,y, y' \right) \) is called linear.
For example, the equation \( y'' = \sin x + 3 y^2 + 2 \left( y' \right)^2 \) is a second order nonlinear differential equation, while the equation \( y'' = (\cos x)\,y \) is a linear one.
A function \( y=\phi (x) \) is a solution of \( y'' = F\left( x,y, y' \right) \) in some open interval \( a < x < b \) (perhaps infinite), having derivatives up to the second order throughout the interval, if φ(x) satisfies the differential equation in an open interval (𝑎, b), that is,
For many of the differential equations to be considered, it will be found that solutions of \( y'' = F\left( x,y, y' \right) \) can be included in one formula, either explicit
Second order differential equations are widely used in science and engineering to model real world problems. The most famous second order differential equation is Newton's second law of motion, \( m\,\ddot{y} = F\left( t, y, \dot{y} \right) ,\) which describes a one-dimensional motion of a particle of mass m moving under the influence of a force F. In this equation, y=y(t) is the position of a particle at a time t, \( \dot{y} = {\text d}y/{\text d}t \) is its velocity, \( \ddot{y} = {\text d}^2 y/{\text d}t^2 \) is its acceleration, and F is the total force acting on the particle.
For given two numbers y_{0} and y_{1}, we impose the initial conditions on y(x) in the form
Since the differential operator \( \texttt{D} = {\text d}/{\text d}x \) is an unbounded operator, it is convenient to rewrite the initial value problem in equivalent integral form:
The general linear differential equation of the second order is an equation that can be written as
The points where the coefficients of \( y'' (x) + p (x) \,y' (x) + q (x)\, y(x) = f(x) \) are discontinuous or undefined are called the singular points of the equation. These points are usually not used in the initial conditions except some cases. For example, the equation \( (x^2 -1)\,y'' + y=1 \) has two singular points x=1 and x=-1 that must be excluded. If in opposite, the initial condition \( y(1) =y_0 \) is imposed, then the differential equation dictates that \( y_0 =1 ; \) otherwise, it has no solution.
Equation \( y'' + p(x)y' + q(x) y =f(x) \) is a particular case of the general linear differential equation of the n-th order
Solution: To determine the validity interval (= the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution), we divide both sides of the differential equation by \( x(x^2 -4)=x(x-2)(x+2) \) to obtain \( y'' + p(x)y' + q(x)y =f(x) \) or
soln = DSolve[{x''[t] + 25 x[t] == 0, x[0] == 1, x'[0] == 0}, x[t], t]
Plot[x[t] /. soln, {t, -1, 2.5}]
s[t_] = x[t]/.soln[[1]]
Plot[s[t],{t,0,3},AxesLabel->{"t","Displacement"}]
s1[t_] = x[t] /. soln[[1]]
>>>> displace1.jpg
When does the maximum excursion occur?
What is the maximum excursion?
soln = DSolve[{x''[t] + 2 x'[t] + 37 x[t] == 0, x[0] == -1,
x'[0] == 5}, x[t], t]
s3[t_] = x[t] /. soln[[1]]
Plot[s3[t], {t, 0, 3}, PlotRange -> {-1, 1}]
Out[20]= -(1/3) E^-t (3 Cos[6 t] - 2 Sin[6 t])
Out[21]=
>>>>> solution3.jpg
s3[t] /. c3_ Exp[c4_] (c1_ Cos[a_] + c2_ Sin[a_]) -> c3*Sqrt[c1^2 + c2^2]
>>>>> amplitude.jpg
Forced Oscillations
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