Differential equation: An equation containing an independent variable, dependent variable and differential coefficient of dependent variable with respect to independent variable is called a differential equation.Order of a differential equation: The order of a differential Equation is the order of the highest order derivative appearing in the equation. For example, in the equation , the Read more about Ordinary D.F equations, their order and degree[…]

Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by   for all x ϵ [a, b] is called the integral function of the function f(x). Property I: The integral function of an integrable function is always continuous. Property II:  if φ(x) is differentiable on (a, b) and φ’(x) Read more about Integral Function and its Properties[…]

We have the following methods of integration:  (i) Integration by substitution or change of independent of independent variable.  (ii) Integration by parts.  (iii) Integration of rotational algebraic function by using partial fractions. Integration by substitution: If φ (x) is a continuous differentiable function, then to evaluate integrals of the form ∫f (φ(x)) φ’ (x) dx, Read more about Methods of Integration[…]

Primitive or anti-derivative of a function: A function ø is called a primitive or an ant derivative of a function f(x) if ø’(x) = f(x). Let ø (x) be a primitive of a function f(x) and let C be any constant then, d/dx [ø (x) + C] = ø’ (x) = f (x) [∵ø’ (x) Read more about Integral as Anti Derivative[…]

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Terms Related to an Ellipse: Fundamental Terms Ellipse (Horizontal ellipse) Conjugate Ellipse(Vertical ellipse) (a) Equation x²/a² + y²/b² = 1 (a > b) x²/a² + y²/b² = -1 (a < b) (b) Graph (c) Centre C (0, 0) C (0, 0) (d) Vertices (±a, 0) (0, ±b) (e) Length of major axis 2a 2b (f) Read more about Ellipse[…]

Let S be the focus, ZK the directrix and e the eccentricity of the ellipse whose equation is required. Draw SK perpendicular from S on the directrix. Divide SK internally and externally at A and A’ (on SK produced) respectively in the ratio e : 1. ∴ ⇒ SA = e. AK … (1) And Read more about Standard form of the ellipse[…]

The locus of the mid-points of a system of parallel chords of a conic is known as its diameter. Equation of diameter of a parabola: Let y = mx + c be a system of parallel chords of the parabola y² = 4ax.Here, m is a constant and c is a variable. The line y Read more about Diameter[…]

ocal chord: A chord of parabola is a focal chord, if it passes through focus Some other standard forms of parabola: A. y² = – 4ax Co – ordinates of vertex = (a, 0) Co – ordinates of focus = (-a, 0) Equation of directrix = x – a = 0 Equation of axis is Read more about Standard form of Parabola and its Various Forms II[…]

Equation of parabola in its standard form: Let S be focus and ZZ’ be directrix of a parabola. Draw SK perpendicular from S on directrix and bisect SK at A ⇒ AS = SK Where A lies on parabola SK = 2a ⇒ SA = SK = a Let choose A be origin, AS as Read more about Standard form of Parabola and its various forms[…]