Differential Equations Overview

Classification By Degree

  • Degree of a Differential Equation (D.E) is determined by the highest derivative's power in the equation.
  • Examples:
    • For the equation 3y2(dydx)3−dydx=sin⁡(x2)3y^2 (\frac{dy}{dx})^3 - \frac{dy}{dx} = \sin(x^2):
    • Highest order derivative = 2 (order)
    • Degree = 1 (power of the highest derivative)
    • For d2ydx2+sin⁡(2y)+3x=0\frac{d^2y}{dx^2} + \sin(2y) + 3x = 0:
    • Degree is NOT DEFINED (due to sin function)

Classification By Linearity

  • An nth order ODE is linear if it can be expressed as:
    F(x,y,y′,y′′,…,y(n))=0F(x,y,y',y'',…,y^{(n)}) = 0
  • Normal form: dydx=f(x,y,y′,y′′,…,y(n−1))\frac{dy}{dx} = f(x,y,y',y'',…,y^{(n-1)})
  • Linear ODE forms:
    1. 1st order: a1(x)dydx+q(x)y=g(x)a_1(x)\frac{dy}{dx} + q(x)y = g(x)
    2. 2nd order: a(x)d2ydx2+b(x)dydx+c(x)y=g(x)a(x)\frac{d^2y}{dx^2} + b(x)\frac{dy}{dx} + c(x)y = g(x)
  • Non-linear ODE involves nonlinear functions of y or its derivatives (e.g., sin⁡(y),ey\sin(y), e^y).

Summary of Linear and Non-Linear D.E.

  • A D.E is linear if:
    • Dependent variable and its derivatives are in the first power only.
    • No products of dependent variables with their derivatives.
    • No non-linear functions (e.g., sin⁡(y),ey\sin(y), e^y).

Formation of Differential Equations

  • To form a D.E from an ordinary equation, differentiate and eliminate arbitrary constants.
    • Example with y = Ax + x² leads to a D.E of order 1 after eliminating A.
  • Continuing differentiation helps eliminate more constants.
  • The general solution contains as many arbitrary constants as the order of the D.E.

Solution of Differential Equations

  • A solution is an expression for the dependent variable satisfying the D.E.
  • General solution encompasses all possible solutions with arbitrary constants.
  • Particular solutions arise when specific values are assigned to the constants.

Verification of Solutions

  • To verify, substitute the proposed solution back into the original D.E. to check if LHS = RHS.

Autonomous Differential Equations

  • An ordinary D.E is autonomous if the independent variable does not explicitly appear, i.e., can be expressed as: dydt=f(y)\frac{dy}{dt} = f(y).
  • Critical points are the zeros of f(y).

Separable Equations

  • An equation can be written as h(y)dydx=g(x)h(y)\frac{dy}{dx} = g(x).
  • This form allows integration of both sides separately for solving.

Example Problem

  • Solve the initial value problem: dydx=−1;y(4)=−3\frac{dy}{dx} = -1; y(4) = -3:
    • Leads to a circular equation upon integration and substitution of initial conditions.