Differential Equations — NDA Mathematics

Formulas (1)

• Order and degree of a differential equation · Order and degree
  $$\text{order} = \text{order of the highest derivative present} \qquad \text{degree} = \text{power of the highest-order derivative, after making it polynomial in the derivatives}$$

Watch out for (3)

• Clear fractional powers BEFORE reading the degree
  → Order and degree of a differential equation
• Degree is undefined when a derivative sits inside a transcendental
  → Order and degree of a differential equation
• Count INDEPENDENT constants
  → Solutions and arbitrary constants

Watch out for (3)

• Differentiate as many times as there are constants
  → Eliminating arbitrary constants
• The order of the resulting ODE equals the number of constants
  → Eliminating arbitrary constants
• A circle needs equal squared-term coefficients
  → Matching an ODE to its general solution

Formulas (4)

• Separation of variables · Separation of variables
  $$\int \frac{dy}{g(y)} = \int f(x)\,dx + C$$
• Reducible to separable by substitution · Exact differentials to recognise
  $$x\,dy + y\,dx = d(xy) \qquad \frac{x\,dy - y\,dx}{y^2} = d\!\left(\frac{x}{y}\right) \qquad x\,dx + y\,dy = \tfrac12\,d(x^2+y^2)$$
• Linear equations and the integrating factor · Integrating factor
  $$\mu = e^{\int P(x)\,dx},\qquad \frac{d}{dx}(\mu y) = \mu Q,\qquad y\,\mu = \int Q\,\mu\,dx + c,\qquad \frac{dx}{dy}+P(y)\,x=Q(y):\ \mu=e^{\int P(y)\,dy},\qquad \frac{dy}{dx}+Py=Qy^{n}:\ v=y^{1-n}$$
• Initial-value problems and growth/decay · Growth/decay and order
  $$\frac{dN}{dt} = kN \;\Rightarrow\; N = N_0 e^{kt} \qquad \text{number of arbitrary constants} = \text{order}$$

Watch out for (7)

• Take logs / exponentials to unlock separation
  → Separation of variables
• One arbitrary constant only — and add it at the integration step
  → Separation of variables
• Spot the glued combination first
  → Reducible to separable by substitution
• Memorise the exact differentials with the right sign
  → Reducible to separable by substitution
• Put the equation in STANDARD form before reading off P
  → Linear equations and the integrating factor
• Bernoulli substitution is $v=y^{1-n}$, not $y^{n-1}$
  → Linear equations and the integrating factor
• Apply the initial condition to the GENERAL solution
  → Initial-value problems and growth/decay