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

\[
z p^{2}-y^{2} p+y^{2} q=0
\]