Differential Equations and Vector Calculus – Key Vocabulary

Page 1

• Overview of the syllabus (5 units)
  • Unit-1 – Ordinary Differential Equations (ODE): higher-order linear equations, Bernoulli, Exact, operators $D$, complementary function (C.F), particular integral (P.I), variation of parameters.
  • Unit-2 – Equations reducible to linear form, Cauchy & Legendre equations, simultaneous linear ODEs with constant coefficients.
  • Unit-3 – First–order Partial Differential Equations (PDE): homogeneous & non-homogeneous forms, higher-order PDEs.
  • Unit-4 – Multivariable calculus I: vector differentiation (gradient, divergence, curl, vector identities).
  • Unit-5 – Multivariable calculus II: vector integration (line, surface & volume integrals), Green, Stokes & Divergence theorems.

Page 2

• Definition of a differential equation.
  • Ordinary vs. Partial DE.
• Examples that distinguish ODE ($y = y(x)$) from PDE ($u = u(x,y)$).

Page 3

• Order = highest derivative present; degree = power of that derivative once the equation is polynomial in derivatives.
• Worked examples:
  • $\frac{d^2y}{dx^2}+7x=0$ → order 2, degree 1.
  • $y=x\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^2$ → order 1, degree 2.

Page 4

• Formation of a DE by eliminating constants from $y = \cos(n x + \beta)$, $y = Ax + A^2$, $y = Ae^{x}+Be^{-x}$.
• Resulting second-order DEs shown.

Page 5 – Page 6

• More elimination practice producing third-order equations from $y=Ax^3+Bx^2$ and $y=e^x(A\cos x+B\sin x)$.

Page 7

• Differential equation of the family of circles $(x-h)^2+(y-k)^2=a^2$ derived: $(y-k)y''=(1+y'^2).$

Page 8

• Variable-separable concept: rearrange to $f(x)dx = g(y)dy$ then integrate.
• Example manipulation leading to $\log\lvert y\rvert = \log\lvert x\rvert +C.$

Page 9

• Homogeneous first-order ODE ($dy/dx = F(y/x)$) solved by substitution $y=vx$.
• Trig example giving $\tan^{-1}y - \tan^{-1}v = C.$

Page 10

• Definitions: general solution (contains as many arbitrary constants as its order), particular solution (constants assigned specific values).
• Four standard first-order methods: separable, linear, Bernoulli, exact.

Page 11

• Recap of separable type with worked exponential example producing $3e^{2y}=2e^{3x}+2x+C.$

Page 12

• Trigonometric separable example: $\sec x\tan x\,dx+\sec y\tan y\,dy=0$ leads to $\tan x\tan y=C.$

Page 13

• Partial-fraction decomposition inside separable integrals.
• Logarithmic solution $\log y + y = x + \log x +C.$

Page 14–15

• Additional separable integrals including square-root and exponential substitutions.

Page 15 (Linear first-order)

• Standard form $\frac{dy}{dx}+P(x)y=Q(x)$.
• Integrating factor $IF=e^{\int P\,dx}$.
• General solution: $y(IF)=\int Q(IF)dx + C$.

Page 16–18

• Three linear examples solved:
  1. $(x+1)dy/dx +4e^{3x}=y$ → solution $y(x)=e^{3x}+3x+C(x+1)^{-1}.$
  2. $\cos x\,dy+\tan x\,y=\sin x$ gives $y(\tan x-1)=C.$
  3. $\alpha\log x\,dy+y = \log x$ processed via substitution $t=\log x$.

Page 19–24 (Bernoulli)

• Bernoulli form $dy/dx+Py=Qy^n$. Divide by $y^n$, set $z=y^{1-n}$ to obtain linear DE in $z$.
• A full transformation example yields final explicit $y(x)$.

Page 25–29 (Exact & Integrating Factor)

• Exact test $\partial M/\partial y = \partial N/\partial x$ for $Mdx+Ndy=0$.
• If not exact, find integrating factor by inspection or formula $\mu(x) = e^{\int (N<em>x-M</em>y)/M\,dx}$.
• Solved examples:
  • $(x^2-ay)dx+(y^2-ax)dy=0$.
  • $y(x^2+y^2-a^2)dx + x(x^2-y^2-b^2)dy=0$.

Page 30–32 (IF, non-exact cases)

• Homogeneous type integrating factor $IF=1/(Mx+Ny)$.
• Problems worked with multipliers and partial fractions.

Page 33–37 (Higher-order Linear ODE)

• Operator notation $D=d/dx$.
• Auxiliary (characteristic) equation $f(m)=0$ derived from constant-coefficient linear ODE.
• Four root cases:
  1. Distinct real $m<em>1,m</em>2,\dots$: $CF=\sum C<em>ke^{m</em>kx}$.
  2. Repeated: multiply by powers of $x$.
  3. Complex pair $\alpha\pm i\beta$: $e^{\alpha x}(C<em>1\cos\beta x+C</em>2\sin\beta x)$.
  4. Repeated complex.
• Sample solutions for $(D^2+D-2)y=0$ and $(D^2+6D+9)y=0$.

Page 38–56 (Particular Integral rules)

• Inverse operator method: $PI=\frac{1}{f(D)}X$.
• Special input functions:
  • Exponential $e^{ax}$ ⇒ replace $D$ with $a$.
  • $\sin ax,\cos ax$ ⇒ replace $D^2$ with $-a^2$.
  • Polynomials $x^m$ ⇒ expand $1/f(D)$ in ascending $D$.
  • Products $e^{ax}V(x)$ ⇒ use shift $D→D+a$.
• Extensive worked examples up to third order, partial fractions, repeated roots.

Page 57–60 (ODE examples with trig/exponential forcing)

• Application of all PI rules giving complete solutions combining CF+PI.

Page 61–66 (Variation of Parameters)

• For $y''+P(x)y'+Q(x)y=R(x)$ with two known independent $y<em>1,y</em>2$:
  $u<em>1'=-\frac{y</em>2R}{W},\qquad u<em>2'=\frac{y</em>1R}{W},\qquad W=y<em>1y</em>2'-y<em>1'y</em>2.$
• Examples include $y''-4y'+4y=8x^2e^{2x}\sin ax$ where $y<em>1=e^{2x}, y</em>2=xe^{2x}.$

Page 67–70 (Cauchy & Legendre Equations)

• Cauchy form $x^n y^{(n)}+\dots=F(x)$; use substitution $x=e^t$ so that $D<em>t = xD</em>x$.
• Example: $x^2y''-4xy'+6y = x^2$ mapped to constant-coefficient $D(D-1)-4D+6$.

Page 71–78 (First-order PDE)

• Standard notations $p=\partial z/\partial x,\;q=\partial z/\partial y$.
• Formation by eliminating constants; family of circles example reproduced.
• Four standard forms:
  1. $f(p,q)=0$ ⇒ complete integral $z=ax+by+c$ constrained by $f(a,b)=0$.
  2. $f(x,p)=g(y,q)$ ⇒ integrate separately: $dz = P(x)dx + Q(y)dy$.
  3. Clairaut $z=px+qy+f(p,q)$ ⇒ replace $p, q$ with $a,b$ then envelope gives singular solution.
  4. General Lagrange $Pp+Qq=R$. Subsidiary (auxiliary) equations: $\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$ solved by grouping or multipliers.
• Charpit method introduced for non-linear first-order PDEs.

Page 79–84 (Vector Differentiation)

• Scalar field $\phi(x,y,z)$ ⇒ gradient $\nabla\phi$ (vector).
• Vector field $\mathbf F=F<em>1\mathbf i+F</em>2\mathbf j+F_3\mathbf k$.
  • Divergence $\nabla\cdot\mathbf F = \partial F<em>1/\partial x + \partial F</em>2/\partial y + \partial F_3/\partial z$ (scalar).
  • Curl $\nabla\times\mathbf F$ (vector, determinant mnemonic).
• Solenoidal: $\nabla\cdot\mathbf F=0$. Irrotational: $\nabla\times\mathbf F=0$.
• Directional derivative in direction $\hat a$: $D_{\hat a}\phi = \nabla\phi\cdot\hat a$.
• Unit normal to surface $\phi=0$ is $\frac{\nabla\phi}{|\nabla\phi|}$.
• Worked gradient, divergence, curl and directional-derivative examples.

Page 85–88 (Vector Integration)

• Differential displacement $d\mathbf r = dx\mathbf i+dy\mathbf j+dz\mathbf k$.
• Line integral $\int_C \mathbf F\cdot d\mathbf r$ (work, circulation).
• Surface integral $\iint<em>S \mathbf F\cdot\hat n\,dS$ with projections $dS=\sqrt{1+(\phi</em>x)^2+(\phi_y)^2}\,dxdy$.
• Volume integral $\iiint_V G\,dV$.

Page 89–95 (Green, Stokes, Divergence)

• Green’s theorem (plane): $\oint<em>C (M\,dx+N\,dy)=\iint</em>R\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy$.
• Stokes’ theorem: $\oint<em>C \mathbf F\cdot d\mathbf r = \iint</em>S (\nabla\times\mathbf F)\cdot\hat n\,dS$.
• Divergence (Gauss) theorem: $\iint<em>S \mathbf F\cdot\hat n\,dS = \iiint</em>V (\nabla\cdot\mathbf F)\,dV$.
• Each theorem verified on rectangular regions, planes & parallelepipeds with explicit calculations.

Page 96 – End (Misc. Worked Problems)

• Multiple integrals evaluated along parametric paths $x=t^2, y=t, z=3t$.
• Use of projections onto coordinate planes to compute surface integrals for plane $2x+3y+6z=12$ inside first octant.
• Combined application of Gauss theorem across six faces of rectangular box to show $\iiint_V \nabla\cdot\mathbf F\,dV = abc(a+b+c)$ for $F=(x-yz,\,y-zx,\,z-xy).$

These page-wise notes capture every principal definition, algorithm, theorem, worked example, and formula mentioned in the transcript, formatted for rapid exam review.

To find the particular integral (PI) for a linear ordinary differential equation with constant coefficients, $f(D)y = X$, the method depends on the form of $X$ (the input function):

• If $X = e^{ax}$ (Exponential function): You replace $D$ with $a$ in the inverse operator $rac{1}{f(D)}$. That is, PI = rac{1}{f(a)}e^{ax}, provided that $f(a) <br /> eq 0$. If $f(a)=0$, a special rule involving multiplication by $x$ and differentiation of $f(D)$ is applied.
• If $X = ext{sin}(ax)$ or $ext{cos}(ax)$ (Trigonometric function): You replace $D^2$ with $-a^2$ in the inverse operator $rac{1}{f(D)}$. That is, PI = rac{1}{f(D^2)} ext{sin}(ax) = rac{1}{f(-a^2)} ext{sin}(ax), provided that $f(-a^2) <br /> eq 0$. Similar rules apply if the denominator becomes zero.
• If $X = x^m$ (Polynomial): You expand $rac{1}{f(D)}$ in ascending powers of $D$ using binomial expansion or long division, and then apply the operators to $x^m$.
• If $X = e^{ax}V(x)$ (Product of exponential and another function): You use the shift rule: PI = e^{ax}rac{1}{f(D+a)}V(x). You then proceed to find the PI for $rac{1}{f(D+a)}V(x)$ based on the type of $V(x)$.

To find the Complementary Function (C.F) of a higher-order linear ordinary differential equation with constant coefficients, you first derive the Auxiliary (characteristic) equation $f(m)=0$. The form of the C.F then depends on the roots of this auxiliary equation:

1. Distinct real roots ($m<em>1, m</em>2, ext{…}$): The C.F is the sum of exponential terms, i.e., $CF=\sum C<em>ke^{m</em>kx}$.
2. Repeated real roots: If a root $m$ is repeated $n$ times, the C.F terms associated with it will include powers of $x$, for example, $(C<em>1+C</em>2x+ ext{…}+C_nx^{n-1})e^{mx}$.
3. Complex conjugate roots ($\alpha\pm i\beta$): For each pair of complex roots, the C.F term is $e^{\alpha x}(C<em>1\cos\beta x+C</em>2\sin\beta x)$.
4. Repeated complex conjugate roots: If a complex pair $\alpha\pm i\beta$ is repeated, similar to repeated real roots, you multiply by powers of $x$, for example, $e^{\alpha x}((C<em>1+C</em>2x)\cos\beta x+(C<em>3+C</em>4x)\sin\beta x)$ for a pair repeated twice.