Solve your IGNOU Doubts

Solve your IGNOU Doubts

Question:

a) Solve the following DEs

(i) $\left ( \frac{dy}{dx}-1 \right )^2\left ( \frac{d^2y}{dx2}+1 \right )^2\,y=sin^2(x/2)+e^x+x.$ (ii) $2x^2y\left ( \frac{d^2y}{dx^2} \right )+4y^2=x^2\left ( \frac{dy}{dx} \right )^2+2xy\left ( \frac{dy}{dx} \right ).$

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Question:

c) Solve: $x\frac{dy}{dx}+$ In $y=x\,y\,e^x.$

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Question:

Show that $\sum_{n=1}^{\infty }(-1)^{n+1}\frac{5}{7n+2}$ is conditionally convergent.

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Question:

b) Find the charge on the capacitor in an RLC circuit at $t=0.0$ sec.when $L=0.05$ Henry, $R=2$ ohms, $C=0.01$ Farad. $E(t)=0,q(0)=5$ Columbus and $i(0)=0.$

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Question:

Test the following series for convergence,

(i) $\sum_{n=1}^{\infty }n\,x^{n-1},x>0.$

(ii) $\sum_{n=1}^{\infty }\left [ \sqrt{n^4+9}-\sqrt{n^4-9} \right ]$

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Question:

a) Solve: $\frac{d^2y}{dx^2}-2$ tan $x\frac{dy}{dx}+5y=e^x$ sec $x.$

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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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