Ordinary Differential Equations: Part 1 2


A differential equation is an equation, where the unknown is a function and both the function and its derivatives may appear in the equation.

Let us first look at some famous laws in the history of science:

  • Newton’s law: Mass times acceleration equals force, ma=f, where m is the particle mass, a=\frac{d^2x}{dt^2} is the particle acceleration, and f is the force acting on the particle. Hence Newton’s law is the differential equation :

m\frac{d^2x}{dt^2}=f(t,x(t),\frac{dx}{dt}(t))
where the unknown is x(t)—the position of the particle in space at the time t. As we
see above, the force may depend on time, on the particle position in space, and on the
particle velocity.

Note: This is a second order Ordinary Differential Equation (ODE).

  • Radioactive Decay: The amount u of a radioactive material changes in time as follows,
    \frac{du}{dt}=-ku(t) \textrm{ } k \gt 0
    where k is a positive constant representing radioactive properties of the material.

Note: This is a first order ODE.

  • Schrödinger Equation (Wave Equation): The Schrodinger Equation is:

\frac{\partial^2\psi}{\partial x^2} + \frac{8\pi^2m}{h^2}(E-V)\psi=0

where,

  1. x:Position
  2. \psi: Schrödinger Wave Function
  3. E: Energy
  4. V: Potential Energy

Note: This is a second order in time and space Partial Differential Equation (PDE).

There are numerous other applications of Differential Equations in varied fields including, but not limited to:

Hence, we can see that differential equations are essential for a mathematical description of nature, and they lie at the core of many theories, and is a very essential tool to any scientist in any field of study.

Definitions:

Let us get comfortable with some basic definitions:

  •  Ordinary Differential Equations (ODE)— the unknown function depends on a single independent variable.
  • Partial Differential Equations (PDE)—The unknown function depends on two or more independent variables, and their partial derivatives appear in the equations.
  • The order of a differential equation is the highest derivative order that appears in the equation.
  • A first order ODE on the unknown y is y'(t)=f(t,y(t))where f is given and y'=\frac{dy}{dt}.

Linear Differential Equations:

The First Order ODE is linear iff the source function f is linear on its second argument,

y'=a(t)y+b(t)

The linear equation has constant coefficients iff both a and b above are constants. Otherwise
the equation has variable coefficients.

Example:

  • y'=y-32 —- First Order Linear ODE with constant coefficients.
  • y'=\frac{y}{t}-3t —-First order linear ODE with variable coefficients.
  • y'=-\frac{7}{y^2}+4t^3 —– Non-Linear

A real-valued function y\textrm{ } : D \subset \mathbb{R} \rightarrow \mathbb{R} defined on a domain D is said to be the solution of a differential equation iff the equation is satisfied for all values of the independent variable, say t, on the domain D.

Theorem 1 (Constant Coefficients): The linear differential equation

y'=ay+b

with a \neq 0, b constants, has infinitely many solutions,

y(t)=ce^{at}-\frac{b}{a} \textrm{ } c \in \mathbb{R}

This is called the general solution of the differential equations. The constant c can be obtained using some given initial values.

Proof: Try the proof. Hint:

  • First prove for b=0.
  • Then take y_{new}=y+\frac{b}{a}.

Method 2: Integrating Factor Method:

The theorem given above cannot be used for all linear equations with variable coefficients. However, we can solve those by using a method called Integrating Factor Method.

  • Write the equation in standard form: \frac{dy}{dt}+a(t)y=b(t).
  • Determine the integrating factor. \textrm{Integrating Factor } = e^{\int a(t)dt}
  • Multiply the equation in standard form by the integrating factor.
  • Using the product and chain rule of differentiation, write the left hand side of the equation in the following way:e^{\int a(t)dt}(\frac{dy}{dt}+a(t)y)=\frac{d}{dt}(e^{\int a(t)dt}y)
  • Hence, we can see:  \frac{d}{dt}(e^{\int a(t)dt}y)=b(t)e^{\int a(t)dt}
  • Integrate both sides of the new equation:  \int \frac{d}{dt}(e^{\int a(t)dt}y) dt=\int b(t)e^{\int a(t)dt} dt
    The fundamental theorem of Calculus gives: \int \frac{d}{dt}(e^{\int a(t)dt}y) dt=e^{\int a(t)dt}y +C_1, where C_1 is an arbitrary constant. Let solution of the other equation is: B(t)+C_2. Then: e^{\int a(t)dt}=B(t)+C_3
  • Divide by integrating factor to get the solution:y=B(t)e^{-\int a(t)dt}+C_3 e^{-\int a(t)dt}

Hence, we can now give Theorem 2:

Theorem 2 (Variable Coefficients): If the functions a, b are continuous, then y'=a(t)y+b(t) has infinitely many solutions given by:

y(t)=ce^{A(t)}+e^{A(t)}\int e^{-A(t)}b(t)dt

where A(t)=\int a(t)dt and c \in \mathbb{R}.

Thus we have successfully solved the Linear Differential Equations for both constant and variable coefficients.

Bernoulli Equation

Definition: The Bernoulli Equation is:

y'=p(t)y+q(t)y^n

where p, q are given functions and n \in \mathbb{R}.

Remarks:

  • For n 6= 0, 1 the equation is nonlinear.
  • If n = 2 we get the logistic equation.
  • This is a non-linear equation which can be transformed into a linear one.

Theorem 3: The function y is a solution of the Bernoulli Equation:

y'=p(t)y+q(t)y^n \textrm{ } n \neq 1

iff the function v=\frac{1}{y^{n-1}} is solution of the linear differential equation

v'=-(n-1)p(t)v-(n-1)q(t)

Separable Equations:

More often than not nonlinear differential equations are harder to solve than linear equations. Separable equations are an exception—they can be solved just by integrating on both sides of the differential equation.

Definition: A separable differential equation for a function y is

h(y)y'=g(t)

where h and g are given functions.

Theorem 4: If h, g are continuous, with h \neq 0, then h(y)y'=g(t) has infinitely many solutions y satisfying the algebraic equation:H(y(t))=G(t)+c

where c \in \mathbb{R} is arbitrary, H and G are antiderivatives of h and g.

Thus, in this article, we have learned the methodology for solving Linear Differential Equations, with both constant and varying coefficients, Bernoulli Equations, and Separable Equations. We will continue in the next post, starting with the Euler Homogenous Equations.

 


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