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Prove that \[ \int x^4 J_1 (x)dx = x^4 J_2(x)-2x^3J_3 (x) +c \]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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520
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Obtain the complex from of Fourier series for f(x)=em in (-I, L)
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Find Laplace transform of \[ i) \ \cosh t \int^1_0 e^u \sinh u \\ ii) \ t\sqrt{1+\sin t} \]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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668
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Find an analytic function f(z)=u+iv where. \[ u-v = \dfrac {x-y}{x^2 + 4xy - y^2} \]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Obtain the Fourier Series expansion for the function \[\begin {align*} f(x)&=1+\dfrac {2x}{\pi}, \ \pi\le x \le 0 \\ &=1-\dfrac {2x}{\pi} , \ 0\le x\le \pi \end{align*} \]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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\[ \text{If } \overline{r}=x\widehat{i}+y\widehat{j}+z\widehat{k} \ \text{prove that, } \nabla r^n=nr^{n-2}\overline{r}\]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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In what direction is the directional derivative of ϕ=x2y2z4 at (3, -1, 2) maximum. Find its magnitude.
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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469
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Prove that \[ J_4 (x) \left ( \dfrac {48}{x^2} - \dfrac {8}{x} \right )J(x)- \left ( \dfrac {24}{x^2}-1 \right )J_0(x) \]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Solve the differential equations using Laplace Transform \[ \dfrac {d^2y}{dt^2} + 2 \dfrac {dy}{dt}+ y = 3te^{-t} \] given y(0)=4 and y'(0)=2.
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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\[ \text {If} \overline {F}(y \sin z-\sin x)\widehat{i}+ (x\sin z+2yz)\widehat{j}+ (xy\cos z+y^2)\widehat{k} \] Show that \(\overline{F}\) is irrotational. Also find its scalar potential.
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Obtain the Fourier Series expression for f(x)=2x-1 in (0,3).
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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519
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Use Stoke's Theorem to evaluate \( \int _c\overline{F}.\overline{dr}\ \text{where}\ \bar{F}=x^2\hat{i}-xy\hat{j} \) and C is the square in the plane z=0 and bounded by x=0, y=0, x=a and y=a.
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Using Convolution theorem find \[L^{-1}\left [ \dfrac{s}{(s^2+1)(s^2+4)} \right ]\]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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348
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Verify Green's Theorem for \( \int _c \overline{F} .\overline{br} \) where \( \bar{F}=(x^2-xy)\hat{i}+(x^2-y^2)\hat{j} \) and C is the curve bounded by x2 = 2y and x=y
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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433
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Prove that \[\int ^b_0 xJ_0(ax)dx=\dfrac{b}{a}J_1(ab)\]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Find the Fourier Series for \( f(x)=\dfrac{3x^2-6\pi x+2\pi^2}{12} \) in (0,2π) Hence deduce that \( \dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}\cdots=\dfrac{\pi^2}{6} \)
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Find an analytic function f(z)=u+iy where. \[u=\dfrac{x}{2}\log(x^2+y^2)-y\tan^{-1}\left ( \dfrac{y}{x} \right )+\sin x \cosh y\]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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420
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if \( \bar{r}=x\hat{i}+y\hat{j}+z\hat{k} \) prove that \( \nabla \log r=\dfrac{\bar{r}}{r^2} \)
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Find the directional derivative of ϕ = 4xz3 - 3x2y2z at (2,-1,2) in the direction of \( 2\hat{i}+3\hat{j}+6\hat{k}. \)
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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978
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Solve the differential equation using Laplace Transform \( \dfrac{d^2y}{dt^2}+4\frac{dy}{dt}+8y=1 \), given y(0)=0 and y'(0)=1
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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If \( \bar{F}=(y^2-z^2+3yz-2x)\hat{i}+(3xz+2xy)\hat{j}+(3xy-2xz+2z)\hat{k}. \) Show that \( \bar{F} \) is irrotational and solenoidal.
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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Obtain the Fourier Series expression for f(x) = 9-x2 in (-3,3)
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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343
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\[If\frac{{\left(a+ib\right)}^{x+1y}\ }{{\left(a-ib\right)}^{x-1y}}=a+i\beta \ then\ find\ \alpha{}\ \&\ \beta{}\]
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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If cos(x+iy) cos(u+iv)=1, where x,y,u, v are real then show that tanh2 cosh2 v=sin2u.
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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if x = u cosv, y = u sinv ( ext {Prove that}dfrac{partial (u,v)}{partial (x,y)},dfrac{partial (x,y)}{partial (u,v)}=1 )
engineering mathematics updated 4.9 years ago by teamques10 ★ 70k
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