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Solution of the differential equation 2x dx – 5y dy = 0 represents
Starting with the differential equation: 2x dx − 5y dy = 0 Separate variables: 2x dx = 5y dy Integrate both sides: ∫ 2x dx = ∫ 5y dy x² = (5/2)y² + C Rearrange the equation: x² − (5/2)y² = C This represents a family of conic sections. Since the equation is of the form: (x²) − (constant)·(y²) = C itRead more
Starting with the differential equation:
2x dx − 5y dy = 0
Separate variables:
2x dx = 5y dy
Integrate both sides:
∫ 2x dx = ∫ 5y dy
x² = (5/2)y² + C
Rearrange the equation:
x² − (5/2)y² = C
This represents a family of conic sections. Since the equation is of the form:
(x²) − (constant)·(y²) = C
it represents a family of hyperbolas (for nonzero C).
Thus, the correct answer is a hyperbola.
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The compound FeS is not attracted by magnet because
(d) FeS does not possess magnetic properties. https://www.tiwariacademy.com/ncert-solutions/class-9/science/chapter-2/
(d) FeS does not possess magnetic properties.
https://www.tiwariacademy.com/ncert-solutions/class-9/science/chapter-2/
See lessIf cos²x is an integrating factor of the differential equation dy/dx + Py = Q, then P can be
Given the differential equation: dy/dx + P y = Q The integrating factor μ(x) is defined by: μ(x) = exp(∫ P dx) We are told that: μ(x) = cos²x Taking the natural logarithm of both sides: ln μ(x) = ln(cos²x) = 2 ln|cos x| Differentiate with respect to x: d/dx [ln μ(x)] = d/dx [2 ln|cos x|] = 2 · (−tanRead more
Given the differential equation:
dy/dx + P y = Q
The integrating factor μ(x) is defined by:
μ(x) = exp(∫ P dx)
We are told that:
μ(x) = cos²x
Taking the natural logarithm of both sides:
ln μ(x) = ln(cos²x) = 2 ln|cos x|
Differentiate with respect to x:
d/dx [ln μ(x)] = d/dx [2 ln|cos x|] = 2 · (−tan x) = −2 tan x
But we also have:
d/dx [ln μ(x)] = P
Thus, we find:
P = −2 tan x
Therefore, the correct answer is:
−2 tan x
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The order of the differential equation of family of parabolas whose axes are parallel to y-axis is
The family of parabolas with axes parallel to the y-axis can be expressed as: y = a x² + b x + c which contains three arbitrary constants, all of them, namely a, b, and c. A differential equation must therefore have a general solution containing three arbitrary constants. Three successive differentiRead more
The family of parabolas with axes parallel to the y-axis can be expressed as:
y = a x² + b x + c
which contains three arbitrary constants, all of them, namely a, b, and c.
A differential equation must therefore have a general solution containing three arbitrary constants. Three successive differentiations eliminate all the arbitrary constants. Indeed three successive differentiations of:
y = a x² + b x + c
After this gives:
y′ = 2a x + b
which twice gives:
y″ = 2a
and thrice gives:
y‴ = 0
So the differential equation that represents the family is:
y‴ = 0
This is a third order differential equation.
Thus, the order of the differential equation is 3.
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The area bounded by the curve y = cosx, x-axis, ordinates x = π/2 and x = π is
The area bounded by the curve y = cos x, the x-axis, and the given ordinates x = π/2 and x = π is given by the definite integral: A = ∫[π/2 to π] cos x dx Evaluating the integral: ∫ cos x dx = sin x Applying the limits: A = sin(π) - sin(π/2) = 0 - 1 = -1 Since area cannot be negative, we take the abRead more
The area bounded by the curve y = cos x, the x-axis, and the given ordinates x = π/2 and x = π is given by the definite integral:
A = ∫[π/2 to π] cos x dx
Evaluating the integral:
∫ cos x dx = sin x
Applying the limits:
A = sin(π) – sin(π/2)
= 0 – 1
= -1
Since area cannot be negative, we take the absolute value:
A = 1 square unit
Thus, the correct answer is: 1 sq. unit
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