Solve your IGNOU Doubts

Question:

Use Cauchy’s mean value theorem to prove that:

$\frac{cos\,\alpha -cos\beta }{sin\alpha sin\beta }=tan\,\theta, 0<\alpha <\theta <\beta <\frac{\pi }{2}$

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Question:

Evaluate: $\lim_{n\rightarrow\infty }\sum_{r=1}^{2n}\frac{n^2}{(2n+r)^3}.$

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Question:

b) Find the temperature in a bar of length l with both ends insulated and with initial temperature in the rod being $sin\frac{\pi x}{l}.$

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Question:

If the partition $P_2$ is a refinement of the partition $P_1$ of $[a,b],$ then $L(P_1,f)\leq L(P_2,f)$ and $U(P_2,f)\leq U(P_1,f).$ Verify this result for the function $f(x)=2cos\,x$ defined over the interval $[0,\frac{\pi }{2}]$ and the partitions $P_1=\left \{0,\frac{\pi }{3},\frac{\pi }{2} \right \}$ and $P_1=\left \{0,\frac{\pi }{6},\frac{\pi }{3},\frac{\pi }{2} \right \}.$

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Question:

a) Using the method of separation of variables, solv $u_{xt}=e^{-t}cos\,x$ when $u(x,0)=0,\frac{\partial u}{\partial t}(0,t)=0.$

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Question:

b) Find the equation of the integral surface of the differential equation

$(x^2-yx)p+(y^2-zx)q=z^2-xy$

which passes through the line $x=1,y=0.$

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Question:

a) Solve the following differential equations

(i) $x^2p+y^2q=(x+y)z.$

(ii) $\sqrt{p-}\sqrt{q+3x}=0.$

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Question:

c) Show that $2z=(ax+y)^2+b,$ where a, b are arbitrary constants is a complete integral of $px+qy-q2=0.$

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Question:

b) Solve the following equation by Jacobi’s method $x^2\frac{\partial u}{\partial x}-\left ( \frac{\partial u}{\partial y} \right )^2-a\left ( \frac{\partial u}{\partial z} \right )^2=0.$

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Question:

a) Verify that the Pfaffian differential equation $yz\,dx\,+(x^2y-zx)dy+(x^2z-xy)dz=0$

is integrable and hence find its integral.

b) Solve the following equation by Jacobi’s method

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Question:

a) Verify that the Pfaffian differential equation $yz\,dx\,+(x^2y-zx)dy+(x^2z-xy)dz=0$

is integrable and hence find its integral.

b) Solve the following equation by Jacobi’s method

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Question:

Show that $R_n(x),$ the lagrange’s form of remainder in the Maclaurin series expansion of $e^{4x}$ , tends to zero as n → ∞.Hence obtain the Maclaurin’s infinite expansion for $e^{4x}.$

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Question:

b) The differential equation of a damped vibrating system under the action of an external periodic force is: $\frac{d^2x}{dt^2}+2m_0\frac{dx}{dt}+n^2x=a$ $\frac{d^2x}{dt^2}+2m_0\frac{dx}{dt}+n^2x=a\,cos\,pt$

Show that, if $n>m_0>0$ the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.

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Question:

Determine the local minimum and local maximum values of the function f defined by $f(x)=3-5x^3+5x^4-x^5.$

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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 $i(0)=0.$