 # IGNOU BMTC-132 Solved Assignment 2023 | B.Sc (G) CBCS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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Details For BMTC-132 Solved Assignment

## IGNOU BMTC-132 Assignment Question Paper 2023

Course Code: BMTC-132

Assignment Code: BMTC-132/TMA/2023

Maximum Marks: 100

\section{Part A (30 Marks)}

1. State whether the following statements are true or false. Justify your answer with the help of a short proof or a counter example:

$$(5 \times 2=10)$$

a) $$y^{2}$$ is an integrating factor of the differential equation:

$6 x y d x+\left(4 y+9 x^{2}\right) d y=0$

b) The solution of the differential equation $$\frac{d y}{d}=y$$ with $$y(0)=0$$ exists, but is not unique.

c) $$\sin \mathrm{x} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}}+\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=0$$ in $$] 0, \pi[$$ is a linear homogeneous equation.

d) The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y} \text { with } \mathrm{y}(0)=0$

exists, but is not unique.

e) The Pfaffian equation $$\left(2 x y^{2}+2 x y+2 x z^{2}+1\right) d x+d y+2 z d z=0$$ is integrable.

2. a) Apply the method of variation of parameter to solve the differential equation:

$y^{\prime \prime}+6 y^{\prime}+9 y=\frac{1}{x^{3}} e^{-3 x}, x>0$

b) Suppose that a thermometer having a reading of $$75^{\circ} \mathrm{F}$$ inside a house is placed outside where the air temperature is $$15^{\circ} \mathrm{F}$$. Two minutes later it is found that the thermometer reading is $$30^{\circ} \mathrm{F}$$. Find the temperature reading $$\mathrm{T}(\mathrm{t})$$ of the thermometer at any time $$\mathrm{t}$$.

3. a) Find the integral surface of the p.d.e.:

$(x-y) p+(y-x-z) q=z$

through the circle $$z=1, x^{2}+y^{2}=1$$.

b) Solve: $$\left(x^{2} y-2 x y^{2}\right) d x-\left(x^{3}-3 x^{2} y\right) d y=0$$.

\section{Part B (40 Marks)}

4. a) Using Charpit’s method, find the complete integral of the p.d.e.:

$2 x z-p x^{2}-2 q x y+p q=0$

b) Using the method of undetermined coefficients, solve the differential equation:

$\left(D^{3}+2 D^{2}-D-2\right) y=e^{x}+x^{2}$

c) Solve the differential equation: $$\frac{\mathrm{dy}}{\mathrm{dx}}=(x+y)^{2}$$.

5. a) A particle falls from rest in a medium in which the resistance is $$\lambda v^{2}$$ per unit mass, $$v$$ being the velocity of the particle at time $$t$$. Prove that the distance fallen in time $$\mathrm{t}$$ is $$\frac{1}{\lambda} \ln \cosh (\mathrm{t} \sqrt{\mathrm{g} \lambda})$$, where $$\mathrm{g}$$ is the acceleration due to gravity.

b) Solve: $$\left(y^{2}+y z\right) d x+\left(z^{2}+z x\right) d y+\left(y^{2}-x y\right) d z=0$$.

6. a) Solve: $$\left(D^{2}+5 D^{\prime}+5 D^{\prime 2}\right) z=x \sin (3 x-2 y)$$.

b) Solve: $$\frac{d x}{y^{2}+y z+z^{2}}=\frac{d y}{z^{2}+z x+x^{2}}=\frac{d z}{x^{2}+x y+y^{2}}$$.

7. a) Using the method of variation of parameters, solve the equation

$\frac{d^{2} y}{d x^{2}}+a^{2} y=\sec a x$

b) Solve $$(p+q)(p x+q y)=1$$, using Charpit’s method.

c) Solve: $$\frac{d y}{d x}=\frac{1}{x+y+1}$$.

\section{Part C (30 Marks)}

8. a) Solve: $$\left(D^{2}-D^{\prime}-2 D\right) z=\sin (3 x+4 y)+e^{2 x+y}$$.

b) Use the method of variation of parameters to solve the following differential equation:

$y^{\prime \prime}-2 y^{\prime}+y=\frac{12 e^{x}}{x^{3}}$

9. a) Solve: $$x^{2} y^{\prime \prime}-2 x y^{\prime}-4 y=x^{2}+2 \ln x$$.

b) Solve the equation $$(7 y-3 x+3) d y+(3 y-7 x+7) d x=0$$.

10. a) Using the method of undetermined coefficients, solve the equation

$\frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+2 y=4 x^{2}$

b) Using Charpit’s method, solve the equation

$z p^{2}-y^{2} p+y^{2} q=0$

$$2\:cos\:\theta \:cos\:\phi =cos\:\left(\theta +\phi \right)+cos\:\left(\theta -\phi \right)$$

## BMTC-132 Sample Solution 2023

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$$cos\:2\theta =cos^2\theta -sin^2\theta$$

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