Sketch the following equations:
I. \(\quad \mathrm{X}(\mathrm{n})=U_{r}(\mathrm{n})-U_{r}(\mathrm{n}-3)-4 \mathrm{U}\)
II. \(\quad X(n)=U(n)-U(n+4)+\partial(n-3)\)
III. \(\mathrm{X}(\mathrm{n})=2^{n}[\mathrm{U}(\mathrm{n})-\mathrm{U}(\mathrm{n}-6)]\)
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Sketch the following equations:I. X(n) = ?? (n) - ?? (n-3) – 4U II. X(n) = U(n)- U(n+4) + ∂ (n-3) III. X(n) = 2 ? [U(n)- U(n-6)]
Transform the following IBVP into a problem with homogeneous boundary conditions.$$ \begin{array}{l} u_{t t}=u_{x x}, \quad 0<x<1, t=>0 \\ a u(0, t)+b u_{x}(0, t)=f_{1}(t) \quad \text { and } \quad c u(1, t)+d u_{x}(1, t)=f_{2}(t), \quad t>0 \\ u(x, 0)=g(x), \quad u_{t}(x, 0)=h(x), \quad 0<x<1 \end{array} $$where \(a, b, c\) and \(d\) are constants.
(25 marks) Solve the following initial value problem using Fourier transform.$$ \begin{array}{l} u_{t}=u_{x x}, \quad-\infty< x <\infty, t= >0 \\ u(x, 0)=\left(1-2 x^{2}\right) e^{-4 x^{2}}, \quad-\infty< x <\infty \end{array} $$with \(u(x, t) \rightarrow 0\) and \(u_{x}(x, t) \rightarrow 0\) as \(x \rightarrow \pm \infty\).
Solve the following IBVP by eigenfunction expansion.$$ u_{t t}=u_{x x}+1+t \cos (\pi x), \quad 0<x<1, quad="" t="">0 $$$$ u_{x}(0, t)=0 \quad \text { and } \quad u_{x}(1, t)=0, \quad t>0 $$$$ u(x, 0)=2 \quad \text { and } \quad u_{t}(x, 0)=-2 \cos (2 \pi x), \quad 0<x<1 $$
(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.$$ \begin{array}{l} u_{t t}=c^{2} u_{x x}, \quad 0< x < \infty, t>0 \\ u(x, 0)=A e^{-\alpha x} \quad \text { and } \quad u_{t}(x, 0)=0, \quad 0< x < \infty \\ u(0, t)=A \cos \omega t, \quad t>0 \\ \lim _{x \rightarrow \infty} u(x, t)=0, \quad \lim _{x...
shown that is a polar equations. r = 4 r. 8 cos e i) sketch two curves on a single polar coordinate system. Shade the region enclosed by r, and ra. ii) Find the two intersection angles between r, and ra iii) Based on the answer i) and ii), find the shaded region's area.
Given the standard enthalpy changes for the following two reactions:(1) \(2 \mathrm{C}(\mathrm{s})+2 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{~g}) \quad \Delta \mathrm{H}^{\circ}=\mathbf{5 2 . 3} \mathrm{kJ}\)(2) \(2 \mathbf{C}(\mathbf{s})+\mathbf{3 H}_{\mathbf{2}}(\mathbf{g}) \longrightarrow \mathbf{C}_{\mathbf{2}} \mathbf{H}_{\mathbf{6}}(\mathbf{g}) \quad \Delta \mathrm{H}^{\circ}=-\mathbf{8 4 . 7} \mathrm{kJ}\)what is the standard enthalpy change for the reaction:(3) \(\mathbf{C}_{\mathbf{2}} \mathbf{H}_{\mathbf{4}}(\mathbf{g})+\mathbf{H}_{\mathbf{2}}(\mathbf{g}) \longrightarrow \mathbf{C}_{\mathbf{2}} \mathbf{H}_{\mathbf{6}}(\mathbf{g}) \quad \Delta \mathrm{H}^{\circ}=?\)Given the standard enthalpy changes for the following two reactions:(1) \(\mathrm{N}_{2}(\mathrm{~g})+2 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{~g}) \quad \Delta \mathrm{H}^{\circ}=66.4 \mathrm{~kJ}\)(2) \(2 \mathbf{N}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow \mathbf{2 N}_{2}(\mathbf{g})+\mathbf{O}_{2}(\mathrm{~g}) \quad \Delta \mathrm{H}^{\circ}=-\mathbf{1 6...
Sketch the following discrete equations. Include 3 non-zero numeric values. n a) x(n) = (5) a(n) b) x(n) = (-+)" u(n) c) x(n) = (2)" u(-n-1)
For the following equations (i) Find the equilibrium points for any le(-00,00). (ii) Sketch the phase line diagram for the indicated. (iii) Find the bifurcation point and sketch the bifurcation diagram. (iv) State if the bifurcation is a saddle-node, transcritical, pitchfork, or none of these. (a) x' = x2 (1+ 24); 1=1, 1=-1.
Find the periodic solutions of the differential equations \((a) \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{ky}=\mathrm{f}(\mathrm{x}),(b) \frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{d}^{3} \mathrm{x}}+\mathrm{ky}=\mathrm{f}(\mathrm{x})\)where \(k\) is a constant and \(f(x)\) is a \(2 \pi\) - periodic function.Consider a Fourier series expansion for \(f(x)\) using the complex form, \(f(x)=\sum_{n=-\infty}^{n=+\infty} f_{n} e^{i n x}\) and try a solution of the form \(y(x)=\sum_{n=-\infty}^{n=+\infty} y_{n} e^{i n x}\)