Detailed Study Notes on Differential Equations and Methods.

Preface

Acknowledgment

Table of Content

1.0 Differential Equation

1.1 Order and Degree of Differential Equation

• Definition: A differential equation contains a differential coefficient. It involves dependent and independent variables.

• Examples:

  • $\frac{dy}{dx} + x^2 y = 0$

  • $\frac{d^2y}{dx^2} + \sin x + y = 0$

  • $\frac{dy}{dx} + \frac{1+x^2}{1-y^2} = 0$

1.2 Formation of Differential Equation

• A differential equation can be formed by differentiating an ordinary equation and eliminating arbitrary constants. The order of the D.E is determined by the number of constants involved.

• Examples:

  • For $y = Ax + A^2$, find $y' = A$.

  • Substitute to find the differential equation from case examples such as $y = Ax^2 + Bx$, etc.

1.3 Separable Equations

• Definition: An equation that can be separated into functions of each variable on either side. The standard form is $\frac{dy}{dx} = f(x)g(y)$.

• Method: Separate variables and integrate both sides:

  1. $f(y) dy = f(x) dx$

  2. Integrate.

1.4 Homogeneous Differential Equations

• Definition: A differential equation is homogeneous if each term is of the same degree.

• Examples: $f(x,y) = x^2 + 3xy + y^2$ is homogeneous.

• Method for Solution:

  • Use substitution $y = vx$ and find $dy/dx$.

1.5 Exact Equations

• Definition: If $M dx + N dy = 0$ is exact, then $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

• Finding Solution:

  • Show that the equation is exact and find specific solutions.

1.6 Integrating Factor

• Definition: An integrating factor converts a non-exact equation into an exact one. The standard form is $\frac{dy}{dx} + P(x)y = Q(x)$.

• Steps to find the integrating factor and solve the differential equation:

  1. Convert to standard form and find the integrating factor $IF = e^{\int P(x) dx}$.

  2. Multiply the equation and integrate both sides.

2.0 Linear Differential Equation of Second Order

2.1 Differential Operator

• Defined by $D = \frac{d}{dx}$.

2.2 Auxiliary Equation with Distinct Roots

• If the auxiliary equation has roots $m<em>1, m</em>2$, solutions can be written as $y = C<em>1 e^{m</em>1 x} + C<em>2 e^{m</em>2 x}$.

2.3 Auxiliary Equation with Repeated Roots

• For repeated roots $m$, general solution is $y = (C<em>1 + C</em>2 x)e^{mx}$.

2.4 Auxiliary Equation with Complex Roots

• For roots in the form $m = a \pm ib$, the complementary function $y = e^{ax}(C<em>1 \cos(bx) + C</em>2 \sin(bx))$.

2.5 Reduction of Variable Order Method

• Method used when one solution is known to find another independent solution.

2.6 Method of Variation of Parameters

• Used to solve non-homogeneous linear differential equations with known solutions from homogeneous equations.

2.7 Power Series Method

• A method to find solutions in form of power series near a point.

2.8 Singular Points of Linear Differential Equations

• Points where coefficients in the differential equation are not analytic, termed as singular points.

3.1 Laplace Transform

• Definition: Convert a function in the time domain into the s-domain.

• Formula: $L[f(t)] = F(s) = \int_0^{\infty} e^{-st} f(t) dt$.

3.2 Laplace Inverse Transform

• Retrieves the original function from its Laplace transform.

3.3 Laplace Transform For Derivatives

• If $f(t)$ is differentiable, then $L[f'(t)] = sL[f(t)] - f(0)$.

4.0 Past Questions and Answers

4.1 2016/2017 Test

• Example Questions on applying different methods to solve differential equations.

4.2 2014/2015 Exams

• Review of past exam questions focusing on key concepts.

4.3 2015/2016 Exams

• Overview of solutions for differential equations.

4.4 2016/2017 Exams

• Collation of significant questions from past papers to consolidate understanding.