Solve your IGNOU Doubts

Question:

c) Solve the following $IVP$ $\frac{d^2y}{dx^2}+\frac{dy}{dx}-2y=-6sin\,2x-18cos2x$

$y(0)=2,y'(0)=2.$

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Question:

Check whether the $\lim_{x\rightarrow0}(x\,cosec\,x)^{x}$ exists or not?

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Question:

b) Solve the equation $\frac{x^2y}{dx^2}+x\frac{dy}{dx}+y=x^m,$ for all positive integer values of m .

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Question:

Verify Bozano–Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval [− ,1 ∞]

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Question:

a) Find the integrating factor of the differential equation

$(6xy-3y^2+y)dx+2(x-y)dy=0$

and hence solve it.

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Question:

b) Solve the following equation by changing the independent variable $(1+x^2)^2y''+2x(1+x^2)y'+4y=0.$

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Question:

Write the inequality 4 ≤ 2x + 3 ≤ 6 in the modulus form.

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Question:

Prove that a strictly decreasing function is always one-one.

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Question:

a) Solve, using the method of variation of parameters $\frac{d^2y}{dx^3}-y=\frac{2}{1+e^x}.$

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Question:

Check whether the intervals ]9,5] and [6,12[ are equivalent or not.

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Question:

Find the following limit.

$\lim_{x\rightarrow 0}\frac{1-cos\,x^2}{x^2\,sin\,x^2}$

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Question:

c) Given that $y_1(x)=x^{-1}$ is one solution of the differential equation $2x^2y''+3xy'-y=0,\,x>0,$

find a second linearly independent solution of the equation.

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Question:

b) Write the ordinary differential equation $y\,dx+(xy+x-3y)dy=0$

in the linear form, and hence find its solution.

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Question:

a) Solve $\frac{dy}{dx}+xy=y^2e^{x2/2}$ sin x.

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Question:

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points: $f(x)=\left\{\begin{matrix} -x^2, &when\,x\leq 0 \\4-5x, &when\,0<x\leq 1 \\ 3x-4x^2, &when\,1<x\leq 2 \\-12x+2x, &when\,x<2 \end{matrix}\right.$

Also check whether the function f is derivable at x = 1.

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Question:

v) The pde $u_{xx+x^2u_{xy}-\left ( \frac{x^2}{2} +\frac{1}{4}\right )u_{yy}=0}$ is hyperbolic in the entire xy-plane.

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Question:

Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]1,0] is an open set.

b) Every bounded sequences is not convergent.

c) The function [:f − 2,2 ] → R defined by $f(x)=\frac{4x+3}{x^{2}+1}$ is uniformly continuous.

d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.

e) The function $f:[0,2]\rightarrow\,R$ defined by $f(x)=x+[x]$ is not integrable.

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Question:

Show that there are infinitely many values of α for which $x^{7}+15x^{2}-30x+\alpha$ is irreducible in $\mathbb{Q}[x].$ .

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Question:

Let $R=\mathbb{Z}[\sqrt{2}]\,and\,M={a+b\sqrt{2}\in R \mid 5\mid\,\,and\,\,5\mid b }.$

i) Show that M is an ideal of R .

ii) Show that if a|5 / or b/|5 , then 5| $(a^{2}+b^{2}),$ for ,a . b ∈ $\mathbb{Z}.$

iii) Hence show that if N is an ideal of R properly containing M , then N = R .

iv) Show that $^R/_M$ is a field, and give two distinct non-zero elements of this field.

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Question:

Let $D=\left \{ f\left ( x,y \right ) +g\left ( x,y \right )i\left | f,g\in \mathbb{Z} \right [x,y]\right \}\subseteq \mathbb{C}[x,y].$ Check whether D is a UFD or not.