# IGNOU MTE-08 Solved Assignment 2023 | MTE

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

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Details For MTE-08 Solved Assignment

## IGNOU MTE-08 Assignment Question Paper 2023

Course Code: MTE-08

Assignment Code: MTE-08/TMA/2023

Maximum Marks: 100

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) If $$\left(\frac{1}{y^{4}}\right)$$ is the integrating factor of the differential equation

$\left(2 x y^{4} e^{y}+2 x y^{3}+y\right) d x+\left(x^{2} y^{2} e^{y}-x^{2} y^{2}-3 x\right) d y=0$

then its solution is $$x^{2} e^{y}+\frac{x^{2}}{y}+\frac{x}{y^{3}}=c$$, where $$c$$ is a constant.

b) Solution of the differential equation

$\frac{\mathrm{d}^{4} \mathrm{y}}{\mathrm{dx}}+\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}}=0$

Satisfying the conditions $$y(0)=y^{\prime}(0)=y^{\prime \prime}(0)=0$$, and $$y^{\prime \prime \prime}(0)=1$$, is $$y=x-\sin x$$.

c) The homogeneous Pfaffian differential equation $$z^{3}\left(x^{2} y-y^{2} z\right) d x+x^{3}\left(y^{2} z-z^{2} x\right)$$ $$d y+y^{3}\left(z^{2} x-x^{2} y\right) d z=0$$ is integrable.

d) Equation $$\sin (x+2 y) p+\cos (2 x-3 y) q=z-\frac{1}{z}$$ is linear.

e) Equation $$(1-y) u_{x x}+2(1-x) u_{x y}+(1+y) u_{y y}+y u_{x}+\mathrm{xu}_{\mathrm{y}}=0$$ is hyperbolic outside the circle $$(x-1)^{2}+y^{2}=1$$

2. a) The rate at which the ice malts is proportional to the amount of ice at the instant. Find the amount of ice left after two hours if half of a quantity melts in 30 minutes.

b) Solve: $$\left(D^{2}-6 D+13\right) y=8 e^{3 x} \sin 4 x+3^{x}$$.

c) The initial value problem $$\frac{d y}{d x}=\frac{2}{x} y, y(0)=0$$ has two solutions $$y=0$$ and $$y=x^{2}$$. Does this result violates existence and uniqueness theorem? Give reasons for your answer.

3. a) Prove that the solutions of the differential equation $$\frac{d y}{d x}-\frac{d x}{d y}=\frac{x}{y}-\frac{y}{x}$$ are given by $$\mathrm{xy}=\mathrm{c}_{1}$$ and $$\left(\mathrm{x}^{2}-\mathrm{y}^{2}\right)=\mathrm{c}_{2}$$, where $$\mathrm{c}_{1}$$ and $$\mathrm{c}_{2}$$ are constants.

b) By changing the dependent variable reduce the following equation to normal form and obtain its solution:

$\frac{d^{2} y}{d x^{2}}-4 x \frac{d y}{d x}+\left(4 x^{2}-1\right) y=-3 e^{x^{2}} \sin 2 x$

4. a) Find the general integral of the equation

$\left(3 z^{2}-2 y z-2 y^{2}\right) p+x(2 y+z) q=x(y-3 z)$

b) Find the complete integral of the partial differential equation:

$\mathrm{z}=\mathrm{px}+\mathrm{q}+\frac{\mathrm{pq}}{\mathrm{y}}$

5. a) Solve the wave equation $$\frac{\partial^{2} u}{\partial \mathrm{t}^{2}}=\mathrm{a}^{2} \frac{\partial^{2} \mathrm{u}}{\partial \mathrm{x}^{2}}$$ under the conditions:

$\begin{gathered} u=0, \text { when } x=0 \text { and } x=\pi, \\ \frac{\partial u}{\partial t}=0, \text { when } t=0 ; u(x, 0)=x, 0<x<\pi . \end{gathered}$

b) Solve:

$\frac{d y}{d x}+\left(\frac{x}{1-x^{2}}\right) y=x \sqrt{y}, y(0)=1$

6. a) Using the method of variation of parameters solve $$\frac{d^{2} y}{d x^{2}}-\mathrm{y}=\frac{2}{1+e^{\mathrm{x}}}$$.

b) What exactly is the advantage of transforming $$f(x, y, z, p, q)=0$$, to $$\mathrm{F}\left(\mathrm{x}, \mathrm{y}, \mathrm{z}, \frac{\partial \mathrm{u}}{\partial \mathrm{x}}, \frac{\partial \mathrm{u}}{\partial \mathrm{y}}, \frac{\partial \mathrm{u}}{\partial \mathrm{z}}\right)=0$$ ? For the equation $$\mathrm{z}+2 \mathrm{u}_{3}-\left(\mathrm{u}_{1}+\mathrm{u}_{2}\right)^{2}=0$$, write down the auxiliary equations.

c) Solve: $$\left(x^{2} D^{2}-y^{2} D^{2}+x D-y D\right) z=6 x y^{2}$$.

7. A square plate $$0 \leq \mathrm{x} \leq 10,0 \leq \mathrm{y} \leq 10$$, has the edges $$\mathrm{x}=0, \mathrm{x}=10, \mathrm{y}=0$$ maintained at zero degree temperature, while the temperature at the edge $$y=10$$ is given by $$20 x-x^{2}$$.

Show that the temperature $$\theta(\mathrm{x}, \mathrm{y})$$, satisfying Laplace’s equation, is given by

$\theta(x, y)=\sum_{m=1}^{\infty} F_{m} \sin \frac{m \pi x}{10}\left(e^{\frac{m \pi}{10} y}-e^{\frac{-m \pi}{10} y}\right)$

where

$\mathrm{F}_{\mathrm{m}}=\frac{1}{\mathrm{~m} \pi \sinh (\mathrm{m} \pi)}\left[200(-1)^{\mathrm{m}+1}+\frac{80}{\mathrm{~m}^{2} \pi^{2}}\left(1-(-1)^{\mathrm{m}}\right)\right]$

8. a) Find the integral surface of the equation $$\left(x^{2}-y z\right) p+\left(y^{2}-z x\right) q=z^{2}-x y$$ passing through the line $$\mathrm{x}=1, \mathrm{y}=0$$.

b) Find the complete integral of $$(p+q)(p x+q y)-1=0$$ by Charpit’s method. 9. a) Find the integral curves of the equations $$\frac{d x}{x^{2}+y^{2}}=\frac{d y}{2 x y}=\frac{d z}{(x+y) z}$$.

b) Solve: $$\left(D^{2}-D^{\prime 2}-3 D+3 D^{\prime}\right) z=e^{x+2 y}$$

10. a) By using method of symbolic operators find the general solution of

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

b) Solve the following differential equation by changing the independent variable,

$x^{2} \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-4 x^{3} y=8 x^{3} \sin x^{2}, x>0$

$$sin^2\left(\frac{\theta }{2}\right)=\frac{1-cos\:\theta }{2}$$

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