Solve your IGNOU Doubts

Question:

if y₁ = 2x + 2 and y₂ = -x²/2 are the solutions of the equation y = xy' + (y')²/2 then are the constant multiples c₁y₁, and c₂y₂, where c₁ and c₂ are arbitrary, also the solutions of the given DE? Is the sum y₁ + y₂ a solution? Justify your answer.

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Question:

Solve the following differential equations:

i) $\frac{dx}{y^2(x-y)}=\frac{dy}{x^2(x-y)}=\frac{dz}{z(x^2+y^2)}$

ii) $(yz+z^2)dx-xzdy+xydz=0$

iii) $(z+z^3)\cos x\,dx-(z+z^3)dy+(1-z^2)(y-\sin x)dz=0.$

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Question:

If ƒ and g are arbitrary functions of their respective arguments, show that $u=f(x-vt+i\alpha y)+g(x-vt+i\alpha y),$ is a solution of $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}$ where, $\alpha^2=1-\frac{v^2}{c^2}.$

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Question:

Reduce the equation x² (y - px) = yp² to clairaut’s form and hence find its complete solution.

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Question:

Find the general solution of the DE

y² ln y = x py + p² Does the equation has any singular solution? If yes, obtain it.

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Question:

Obtain the Riccati equation associated with the equation y" + w²y = 0 and hence find its solution.

b) Solve the differential equation $\frac{d^2y}{dx^2}=a+bx+cx^2$ , given that $\frac{dy}{dx}=0.$ and y =d

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Question:

Show that, for the differential equation

$\frac{d^2y}{dx^2}+a(x)\frac{dy}{dx}+b(x)y=0$

e^mx is a particular integral if m² + am+b = 0. Hence find the value of m so that e^xm is a particular integral of the equation

$(x-2)\frac{d^2y}{dx^2}-(4x-7)\frac{dy}{dx}+(4x-6)y=0$

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Question:

A mass m , free to move along a line is attracted towards a given point on the line with a force proportional to its distance from the given point. If the mass starts from rest at a distance x₀ from the given point, show that the mass moves in a simple harmonic motion.

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Question:

Using the method of variation of parameters, solve the equation

$\frac{d^2y}{dx^2}+y=\operatorname{cosec}x,\quad 0\le x<\frac{\pi}{2}$

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Question:

Classify the following statements as true or false. Give a short proof of a counter example in support of your answer.

i) The solution of the differential equation $\frac{dy}{dx}=y,\quad y(0)=0$ exists, but is not unique.

ii) The differential equation representing all tangents ty = x + t² at the point (t², 2t) to the parabola y² = 4x is x(y')² + yy'+1=0

iii) The p.d.e. au_xx + 2b u_xy + cu,_yy = 0 where a, are constants is irreducible when b² ac = 0

iv) The functions f₁(x) = cos²x, f₂(x) = sin²x, f₃(x) = sec² x and f₄(x) = tan²x are linearly dependent on the interval ] - π/2, π/2 [

v) The solution of the second order partial differential equation $\frac{\partial^2 u}{\partial y\partial x}=x^3-y$ involves two arbitrary constants.

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Question:

Locate and classify the stationary points of the following:

$\text{(i)}\quad f(x,y)=4xy+x^4-y^4$

$\text{(ii)}\quad f(x,y)=xy+\frac{2}{x}+\frac{4}{y},\quad x>0,y>0$

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Question:

Does the function

$f(x,y)=\frac{x^2-y^2}{x^2+y^2},\quad x\ne 0,y\ne 0$ satisfy the requirement of Schwarz’s theorem at (1,1) ? Justify your answer.

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Question:

Identify the level curves of the following functions:

(i) $\sqrt{x^2+y^2}$

(ii) $\sqrt{4-x^2-y^2}$

(iii) x − y

(iv) y / x

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Question:

Suppose S and C are subsets of $\mathbb{R}^{3}$ . S is the unit open sphere with centre at the origin and C is the open cube = $\{P(x,y,z)\mid-1<x<1,-1<y<1,-1<z<1\}.$

Which of the following is true. Justify your answer.

(i) S ⊂ C

(ii) C ⊂ S

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Question:

Find the values of a and b, if

$\lim_{x\rightarrow\infty}\frac{x(1+a\cos x)-b\sin x}{x^3}=1$

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Question:

Use double integration of find the volume of the ellipsoid

$\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1.$

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Question:

Evaluate $\int_C(2x^2+3y^2)dx$ where C is the curve given by

$x(t)=at^2,y(t)=2at,0\le t\le 1.$

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Question:

Find the domain and range of the function f , defined by $f(x,y)=\frac{2xy}{x^2+y^2}.$ Also find two level curves of this function. Give a rough sketch of them.