Solve your IGNOU Doubts

Question:

$a \oplus b = \sqrt{a + b}$ defines a binary operation on $\mathbf{Q}$ , the set of rational numbers.

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Question:

Every integrable function is monotonic.

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Question:

) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .

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Question:

The greatest interger function is continuous on $\mathbf{R}$ .

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Question:

The set $\{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ is an infinite set.

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Question:

. Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate. $\quad (10)$

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Question:

Using Charpit's method, solve the equation

$$zp^2 - y^2p + y^2q = 0.$

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Question:

Using the method of undetermined coefficients, solve the equation

$$\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 4x^2.$

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Question:

) Solve the equation $(7y - 3x + 3)dy + (3y - 7x + 7)dx = 0$ .

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Question:

Solve: $x^2y'' - 2xy' - 4y = x^2 + 2\ln x$ .

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Question:

Use the method of variation of parameters to solve the following differential equation:

$$y'' - 2y' + y = \frac{12e^x}{x^3}.$

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Question:

Solve: $(D^2 - DD' - 2D) z = \sin(3x + 4y) + e^{2x+y}$ .

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Question:

Solve: $\frac{dy}{dx} = \frac{1}{x + y + 1}$ .

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Question:

Solve $(p + q)(px + qy) = 1$ , using Charpit's method.

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Question:

Using the method of variation of parameters, solve the equation

$$\frac{d^2y}{dx^2} + a^2y = \sec ax.$

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Question:

Solve: $\frac{dx}{y^2 + yz + z^2} = \frac{dy}{z^2 + zx + x^2} = \frac{dz}{x^2 + xy + y^2}$ .

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Question:

Solve: $(D^2 + 5DD' + 5D'^2)z = x \sin (3x - 2y)$ .

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Question:

Solve: $(y^2 + yz)dx + (z^2 + zx)dy + (x^2 + xy)dz = 0$ .

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Question:

Solve the differential equation: $\frac{dy}{dx} = (x + y)^2$ . A particle falls from rest in a medium in which the resistance is $\lambda v^2$ per unit mass, v being the velocity of the particle at time t. Prove that the distance fallen in time t is $\frac{1}{\lambda} \ln \cosh (t\sqrt{g\lambda})$ , where g is the acceleration due to gravity.