Solve your IGNOU Doubts

Question:

Show that the equation $x^3+x^2-2x-2=0$ has a real root other than x = − .1

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Question:

b) Show that the wave equation $a^2u_{xx}=u_tt$ can be reduced to the form $u$ =0 by the chang of variable

$=x-at,$ Image ignouassignments-ignouacademy-com--p-doubts-38767 $=x+at.$

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Question:

Using the principle of mathematical induction, show that $1^2+3^2+5x...+(2n-1)^2=\frac{1}{3}n(*4n^2-1)\forall\,n\in N.$

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Question:

Show that $5+\sqrt{2}$ is an algebraic number.

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Question:

Find a and b such that $\lim_{x\rightarrow0}\frac{a\,tan\,x+bx}{x^3}exists.$

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Question:

a) Solve the following differential equations

(i) $[D^3-DD^{'2}-D^2+DD']z=0.$

(ii) $[D^4-D'^{'4}-2D^2\,D'^2]z=0.$

(iii) $[D^2-2DD'^{'4}+D'^2\,]z=12xy.$

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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.