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`int secx/(secx - tanx)dx` equals ______.
Concept: Methods of Integration> Integration by Substitution
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Concept: Properties of Definite Integrals
Evaluate: `int_0^(π/2) sin 2x tan^-1 (sin x) dx`.
Concept: Evaluation of Definite Integrals by Substitution
Using integration find the area of the region {(x, y) : x2+y2⩽ 2ax, y2⩾ ax, x, y ⩾ 0}.
Concept: Area Under Simple Curves
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
`x^2+y^2=4 at (1, sqrt3)`
Concept: Area Under Simple Curves
Find the area bounded by the circle x2 + y2 = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.
Concept: Area Under Simple Curves
Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`
Concept: Solutions of Linear Differential Equation
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Concept: General and Particular Solutions of a Differential Equation
Find the general solution of the following differential equation :
`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`
Concept: General and Particular Solutions of a Differential Equation
Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.
Concept: Formation of a Differential Equation Whose General Solution is Given
Find the integrating factor of the differential equation.
`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`
Concept: Solutions of Linear Differential Equation
Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.
Concept: General and Particular Solutions of a Differential Equation
If y = P eax + Q ebx, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Concept: General and Particular Solutions of a Differential Equation
Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`
Concept: Solutions of Linear Differential Equation
Find the general solution of the following differential equation:
`(dy)/(dx) = e^(x-y) + x^2e^-y`
Concept: Order and Degree of a Differential Equation
Read the following passage:
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An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
Based on the above, answer the following questions:
- Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
- Solve the above equation to find its general solution. (2)
Concept: Homogeneous Differential Equations
Find the projection of the vector `hati+3hatj+7hatk` on the vector `2hati-3hatj+6hatk`
Concept: Multiplication of Vectors >> Scalar (Or Dot) Product of Two Vectors
Vectors `veca,vecb and vecc ` are such that `veca+vecb+vecc=0 and |veca| =3,|vecb|=5 and |vecc|=7 ` Find the angle between `veca and vecb`
Concept: Multiplication of Vectors >> Scalar (Or Dot) Product of Two Vectors
Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1
Concept: Basic Concepts of Vector Algebra
The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A
Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
