Differential Equations

• First-Order ODEs:
  • [[First order ordinary differential equations]], [[existence and uniqueness theorems for initial value problems]].
  • [[Bernoulli’s equation]], [[exact differential equations]], [[integrating factors]], [[orthogonal trajectories]], [[homogeneous differential equations]], [[method of separation of variables]].
• Linear ODEs:
  • [[Linear ordinary differential equations of higher order with constant coefficients]].
  • [[Second order linear ordinary differential equations with variable coefficients]].
  • [[Cauchy-Euler equation]].
  • [Method of variation of parameters].
  • [[Elements of ordinary differential equations]], [[linear differential equations]].
• Systems & Stability:
  • [[Systems of linear first order ordinary differential equations]].
  • [[Planar autonomous systems of ordinary differential equations]]: [[Stability of stationary points for linear systems with constant coefficients]], [[Linearized stability]], [[Lyapunov functions]].
• Special ODE Methods:
  • [[Method of Laplace transforms for solving ordinary differential equations]].
  • [[Series solutions (power series, Frobenius method)]].
  • [[Legendre and Bessel functions and their orthogonal properties]].
  • [[Sturm’s oscillation and separation theorems]], [[Sturm-Liouville eigenvalue problems]].
• Partial Differential Equations:
  • [[Method of characteristics for first order linear and quasilinear partial differential equations]].
  • [[Second order partial differential equations in two independent variables]]: [[classification (PDE)]] and [[canonical forms (PDE)]].
  • [[Method of separation of variables for Laplace equation in Cartesian and polar coordinates]].
  • [[Heat and wave equations in one space variable]].
  • [[Wave equation]]: [[Cauchy problem and d’Alembert formula]], [[domains of dependence and influence]], [[non-homogeneous wave equation]].
  • [[Heat equation]]: [[Cauchy problem]].
  • [[Laplace and Fourier transform methods]].
• Bernoulli’s equation
• exact differential equations
• integrating factors
• orthogonal trajectories
• homogeneous differential equations
• method of separation of variables
• linear differential equations of second order with constant coefficients
• method of variation of parameters
• Cauchy-Euler equation

There are two types of DE: 1. Ordinary DE 2. Partial DE

Order of DE highest derivative Degree of DE power of the highest derivative Formation of DE by differentiating ordinary equation and eliminating constants.

Ordinary Differential Equations

First Order ODE

DE with derivatives w.r.t. one independent variable

Geometric meaning of ODE of 1st order and 1st degree

\displaystyle\frac{dy}{dx}=f(x,y) defines slope fields

???y(x_0)=y_0 gives to point (x_0,y_0) at which the slope field is tangent to the integral curve. Solution of y(x) is the integral curve and it solve the ODE.

Solving ODE of 1st order and 1st degree

DE	Form	Working rule
Variable Separable	f(y)dy=g(x)dx	1. integrate both sides 2. add constant to RHS
Homogeneous	\frac{dy}{dx}=\frac{f(x,y)}{g(x,y)}each term of f(x,y) and g(x,y) is of the same degree	1. put y=vx \frac{dy}{dx}=v+x\frac{dv}{dx} 2. use variable separable method to solve 3. put v=y/x and simplify
Linear	\frac{dy}{dx}+Py=QP,Q are functions of x or constants	1. I.F. = e^{\int P dx} 2. solution is y(I.F.)=\int Q(I.F.)dx
Exact	Mdx+Ndy=0 if \displaystyle\frac{\delta M}{\delta y}=\frac{\delta N}{\delta x} the equation is exact	solution is \int Mdx+\int N^*dy=c where N^* is N without the terms containing x

ODE reducible to homogeneous form

\frac{dy}{dx}=\frac{ax+by+c}{Ax+By+C}

Case	Condition	Working rule
1	\frac{a}{A}\neq\frac{b}{B}	1. substitute, x=X+h, y=Y+k\frac{dy}{dx}\to\frac{dY}{dX}\frac{dY}{dX}=\frac{aX+bY+ah+bk+c}{AX+BY+Ah+Bk+C} 2. solve ah+bk+c=0, Ah+Bk+C=0 and get values of h,k to make the equation homogeneous \frac{dY}{dX}=\frac{aX+bY}{AX+BY}
2	\frac{a}{A}=\frac{b}{B}	1. let \displaystyle\frac{a}{A}=\frac{b}{B}=\frac{1}{m} \frac{dy}{dx}=\frac{ax+by+c}{m(ax+by)+C} 2. put ax+by=z \frac{dy}{dx}=\frac{z+c}{mz+C} 3. find \displaystyle\frac{dz}{dx} and apply variable separable method

(Bernoulli’s Equation) ODE reducible to linear form

\frac{dy}{dx}+Py=Qy^n working rule: 1. divide by y^n and substitute y^{1-n}=z \frac{1}{y^n}\frac{dy}{dx}=\frac{1}{1-n}\frac{dz}{dx} 2. solve \frac{dz}{dx}+P(1-n)z=Q(1-n)

ODE reducible to exact form

Case	Condition	I.F.
1	\frac{\displaystyle\frac{\delta M}{\delta y}-\frac{\delta N}{\delta x}}{N}=f(x)	e^{\int f(x)dx}
2	\frac{\displaystyle\frac{\delta N}{\delta x}-\frac{\delta M}{\delta y}}{M}=f(y)	e^{\int f(y)dy}
3	M=yf_1(xy), N=xf_2(xy)	\frac{1}{Mx-Ny}
4	x^my^n(aydx+bxdy)+x^{m'}y^{n'}(a'ydx+b'xdy)\frac{m+h+1}{a}=\frac{n+k+1}{b}\frac{m'+h+1}{a'}=\frac{n'+k+1}{b'}	x^hy^k
5	Mdx+Ndy=0 is homogeneous and Mx+Ny\neq0	\frac{1}{Mx+Ny}

multiply equation by I.F., it will become exact

ODE of 1st order and higher degree

f(x,y,p)=0,\quad p=\frac{dy}{dx}

Equation	Form	Working rule
solvable for	x	d.w.r.t. y and substitute for p, solve
solvable for	y	d.w.r.t. x and substitute for p, solve
solvable for	p	???
Clairaut’s Equation	y=px+f(p)	y=cx+f(c) c is constant
Singular Solution	f(x,y,p)=0	p.d.w.r.t. p i.e. \frac{\delta f}{\delta p} get p and put in f
Orthogonal Trajectories	f(x,y)=c	trajectory: \frac{\delta f}{\delta x}dx+\frac{\delta f}{\delta y}dy=0 put \displaystyle \frac{dy}{dx}=-\frac{dx}{dy} to get orthogonal trajectory

Linear ODE of order > 1 with constant coefficients

\cdots +ay''+by'+cy=f(x) - non-linear DE is where degree of the dependent variable or derivatives is more than 1

Linear ODE of order 2 with constant coefficients

y''+Py'+Qy=R \iff (D^2+PD+Q)y=R - R is a function of x or constant

Linear independence and dependence

???Let y_1,y_2 be two solutions of two DE, if Ay_1+By_2\neq 0\implies independent, and if =0\implies dependent

Complete Solution = Complementary Function (C.F.) + Particular Integral (P.I.)

Auxiliary equation, A.E. for (D^2+PD+Q)y=R \implies m^2+Pm+Q=0

Nature of roots of A.E.	Roots	C.F.
Real and distinct	m_1,m_2,m_3	c_1e^{m_1x}+c_2e^{m_2x}+\cdots
Repeated	m_1=m_2, m_1=m_2=m_3	(c_1+c_2x)e^{m_1x} (c_1+c_2x+c_3x^2)e^{m_1x}
Complex	m_1=\alpha + i\beta, m_2=\alpha - i\beta	e^{\alpha x}[c_1cos(\beta x) +c_2sin(\beta x)]
Repeated complex	m_1=m_2=\alpha + i\beta, m_3=m_4=\alpha - i\beta	e^{\alpha x}[(c_1+c_2x)cos(\beta x) \quad+(c_3+c_4x)sin(\beta x)]
Irrational	m_1= a+\sqrt{b}, m_2=a-\sqrt{b}	e^{ax}[c_1cosh(\sqrt{b} x) \quad+c_2sinh(\sqrt{b} x)]
Repeated irrational	m_1=m_2= a+\sqrt{b}, m_3=m_4=a-\sqrt{b}	e^{ax}[(c_1+c_2x)cosh(\sqrt{b} x) \quad+(c_3+c_4x)sinh(\sqrt{b} x)]

f(D)y=v	P.I. \displaystyle= \frac{v}{f(D)}
e^{ax}	\frac{e^{ax}}{f(a)} if f(a)=0: \frac{xe^{ax}}{f'(a)} if f'(a)=0: differentiate again and so on
\sin{ax} (or \cos{ax})	put D^2=-a^2, \frac{\sin{ax}}{f(D^2)}=\frac{\sin{ax}}{f(-a^2)} if f(-a^2)=0???
x^n	[f(D)]^{-1}x^n
e^{ax}V	put D\to D+a, \frac{e^{ax}V}{f(D+a)}
xV	\left[x-\frac{f'(D)}{f(D)}\right]\frac{V}{f(D)}
	\frac{V}{D+a}=e^{-ax}\int e^{ax}Vdx

Cauchy-Euler DE

Legendre’s Homogeneous DE

Method of variation of parameters

Partial Differential Equations

[[Syllabus - MATH224.pdf]]

First Order PDE

DE with partial derivatives w.r.t. more than one independent variable.

• Formation of a PDE
• Order and Degree of a PDE
• Linear and Non-linear PDEs of 1st Order
• Complete Solution
• Singular Solution
• General Solution
• Solution of Lagrange’s Linear Equations
• Charpit’s General Method of Solution

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• Linear PDEs of 2nd and higher Orders
• Linear and Non-linear homogeneous and non-homogeneous equations with constant coefficients
• PDEs with variable coefficients reducible to equations with constant coefficients, their complimentary functions and particular Integrals
• Equations reducible to linear equations with constant coefficients

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• Classification of linear PDEs of 2nd order into Hyperbolic, Parabolic and Elliptic types
• Reduction of 2nd order linear PDEs to Canonical (Normal) forms and their solutions
• Solution of linear hyperbolic equations
• Monge’s method for PDEs of 2nd order

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Mathematical models: - [ ] The vibrating string - [ ] Vibrating membrane - [ ] conduction of heat in solids - [ ] the gravitational potential - [ ] conservation laws and the Burgers equation - [ ] Traffic flow

Cauchy problem and wave equations: - [ ] Solutions of homogeneous wave equations with initial boundary-value problems - [ ] and non-homogeneous boundary conditions - [ ] Cauchy problem for non-homogeneous wave equations