Solve your IGNOU Doubts

Question:

Solve the differential equation:

$2x\frac{dy}{dx}+y\left ( 6y^{2}-x-1 \right )=0$

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Question:

If cos and α,cosβ cos γ are the direction cosines of a line, then show that:

$sin^{2}a+sin^{2}\beta +sin^{2\gamma }=2.$

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Question:

Find the limit of:

$f(x,y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$

as $(x,y)\rightarrow (0,0)$ along:

i) $y=3x$

ii) $y=5x$

What can you conclude about $\lim_{(x,y)\rightarrow (0,0)}f(x,y)?Justify your answer.$

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Question:

Find all the first order partial derivatives of the following function:

$h(x,y,t)=e^{x-1}cos(y+t)$

What is the value of $\frac{\partial h}{\partial t}at\left ( 0,\frac{\pi }{2},0 \right )?$

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Question:

Using Charpit’s method, find the complete integral of the differential equation:

$p(1+q^{2})+(b-z)q=0,$

$b$ being a constant.

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Question:

Solve the differential equation:

$x^{2}\frac{d^{2}y}{dx}-4y=x^{2}+2\ In\ x$

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Question:

Show that the differential equation:

$\left ( y^{2} +yx\right )dx+(z^{2}+zx)dy+(xy)dz=0$

is integrable and find its integral.

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Question:

Solve the differential equation:

$\frac{dy}{dx}\frac{-tan\ y}{(1+x)}=(1+x)e^{x}\ sec\ y$

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Question:

Differential equation:

$5x^{2}y^{2}z^{2}=2px^{2}y^{3}+2pz^{2}+9x^{2}y^{2}$

is a semi-linear partial differential equation of first order.

i) The differential equation:

$v\frac{du}{dv}=e^{2v}+uv-u$

has an integrating factor $v\$ exp (−v) .

The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is

$\frac{dx}{y}=\frac{dy}{-w^{2}}=dt$ $with \ y(x_{0})=y_{0.}$

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Question:

The total differential equation corresponding to the family of surfaces , 3 2 x z + x y = c where c is a parameter is $3x^{^{2}}dz+x\left ( y\ dx+xdy \right )=0.$

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Question:

f) Differential equation:

$cos\ x\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+xy^{2}=0$

in $]0,\pi [$ is a linear, homogeneous equation.

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Question:

The differential equation:

$\left [ 1+ \right ({y}')^{2}]={y}'$

is a second order differential equation of degree 3.

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Question:

The unique solution y(x) of an ordinary differential equation:

$\begin{Bmatrix} 0, & for & x& <&0 \\ 1, & for &x &\geq & 0 \end{Bmatrix}$

exists $\forall x\epsilon \mathbb{R}.$

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Question:

The cylindrical coordinates of the point whose spherical coordinates is $\left ( 8,\frac{\frac{\pi }{}}{6} ,\frac{\pi }{2}\right )$ is

$\left ( 8,\frac{\pi }{6} ,0\right )$

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Question:

The function $f(x,y)=$ ln $\left ( \frac{x+yt}{x} \right )$ is not homogeneous function.

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Question:

A real-valued function of three variables which is continuous everywhere is differentiable.

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Question:

From the following figures prepare Trading and Profit and Loss Account of

Lakshmi & Co. for the year ended December 31, 1987.

Rs.

Stock on January 1, 1987	40,000
Purchases	98,000
Commission Received	650
Rent, Rates and Taxes	8,600
Salaries & Wages	12,000
Sales	1,62,100
Returns Inwards	2,400
Returns Outwards	3,000
Sunday Expenses	2,500
Bank Charges	50

Discount Received	750
Carriage on Purchases	2,000
Discount Allowed	530
Carriage on Sales	1,700
Lighting and Heating	2,200
Postage	300
Income from Investments	500
Commission Paid	1,000
Interest paid on a bank loan	550