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Differential Equations
Quiz 1
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Integration factor of differential equation \(\frac{dy}{dx}\) + py = Q, where P and IQ are functions of x is
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∫e
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\(_{e}\)∫pdx
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\(_{e}\)-∫pdx
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None of these
Explanation
None of these
The radius of a circle is increasing at the rate of 0.4 cm/ s. The rate of increasing of its circumference is
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0.4 π cm/s
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0.8 π cm/s
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0.8 cm/s
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None of these
Explanation
0.8 π cm/s
The solution of \(\frac{dy}{dx}\) = 1 + x + y + xy is
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x – y = k(1 + xy)
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log (1 + y) = x + \(\frac{x^2}{2}\) + k
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log (1 + x) + y + \(\frac{y^2}{2}\) = k
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None of these
Explanation
log (1 + y) = x + \(\frac{x^2}{2}\) + k
The degree of the differential equation
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1
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2
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3
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not defined
Explanation
not defined
The degree of differential equation[1 + (\(\frac{dy}{dx}\))²]= \(\frac{d^2y}{dx^2}\) is
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4
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\(\frac{3}{2}\)
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2
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not defined
Explanation
2
The order and degree of the differential equation \(\frac{d^2y}{dx^2}\) + (\(\frac{dy}{dx}\))+ x= 0 respectvely, are
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2 and not defined
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2 and 2
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2 and 3
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3 and 3
Explanation
2 and not defined
If y = e
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\(\frac{d^2y}{dx^2}\) + 2\(\frac{dy}{dx}\) = 0
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\(\frac{d^2y}{dx^2}\) – 2\(\frac{dy}{dx}\) + 2y = 0
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\(\frac{d^2y}{dx^2}\) + 2\(\frac{dy}{dx}\) + 2y = 0
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\(\frac{d^2y}{dx^2}\) + 2y = 0
Explanation
\(\frac{d^2y}{dx^2}\) + 2\(\frac{dy}{dx}\) + 2y = 0
The differential equation for y = A cos αx + B sin αx where A and B are arbitary constants is
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\(\frac{d^2y}{dx^2}\) – α²y = 0
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\(\frac{d^2y}{dx^2}\) + α²y = 0
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\(\frac{d^2y}{dx^2}\) + αy = 0
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\(\frac{d^2y}{dx^2}\) – αy = 0
Explanation
\(\frac{d^2y}{dx^2}\) + α²y = 0
Solution of differential equation xdy – ydx = Q represents
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a rectangular hyperbola
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parabola whose vertex is at origin
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straight line passing through origin
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a circle whose centre is at origin
Explanation
straight line passing through origin
Integrating factor of the differential equation cos x \(\frac{dy}{dx}\) + y sin x = 1 is
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cos x
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tan x
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sec x
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sin x
Explanation
sec x
Family r = Ax + A³ of curves is represented by the differential equation of degree
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1
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2
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3
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4
Explanation
2
Integrating factor of \(\frac{xdy}{dx}\) – y = x – 3x is
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x
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log x
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\(\frac{1}{2}\)
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-x
Explanation
\(\frac{1}{2}\)
Solution of \(\frac{dy}{dx}\) – y = 1 y(0) = 1 is given by
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xy = -e
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xy = -e
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xy = -1
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y = 2e
Explanation
y = 2e – 1
The number of solutions of \(\frac{dy}{dx}\) = \(\frac{y+1}{x-1}\) when y(1) = 2 is
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none
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one
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two
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infinite
Explanation
one
Which of the following is a second order differential equation?
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(y’)² + x = y²
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y’y” + y = sin x
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y” + (y”)² + y = 0
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y’ = y²
Explanation
y’y” + y = sin x
Integrating factor of the differential equation
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-x
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\(\frac{x}{1+x^2}\)
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\(\sqrt{1-x^2}\)
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\(\frac{1}{2}\) log(1 – x²)
Explanation
\(\sqrt{1-x^2}\)
tan x + tan y = c is the general solution of the differential equation
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\(\frac{dy}{dx}\) = \(\frac{1+y^2}{1+x^2}\)
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\(\frac{dy}{dx}\) = \(\frac{1+x^2}{1+y^2}\)
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(1 + x²)dy + (1 + y²)dx = 0
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(1 +x²2)dx+(1 + y²)dy = 0
Explanation
(1 + x²)dy + (1 + y²)dx = 0
The differential equation y \(\frac{dy}{dx}\) + x = c represents
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Family of hyperbolas
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Family of parabolas
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Family of ellipses
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Family of circles
Explanation
Family of circles
The degree of the differential equation \(\frac{d^2y}{dx^2}\) + (\(\frac{dy}{dx}\))³ + 6y = 0 is
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1
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3
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5
Explanation
1
The solution of \(\frac{dy}{dx}\) + y = e, y (0) = 0 is
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y = e(x – 1)
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y = xe
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y = xe+ 1
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y = (x + 1 )e
Explanation
y = xe
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