Advanced Math Archive: Questions from August 17, 2022
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3) [25] Solve the IVP: \( t y^{\prime \prime}-t y^{\prime}+y=2+u_{4}(t) \delta(t-2), y(0)=2, y^{\prime}(0)=-4 \).3 answers -
d) \( y=\frac{e^{2 x}}{1+e^{x}} \) e) \( y=\ln (\sqrt{x-1}) \) f) \( y=\frac{(3 x-3)^{3}}{3 x+3} \) g) \( y=e^{3 x}\left(2 x^{2}+1\right)^{3} \)0 answers -
Calculate \( \frac{d y}{d x} \) for the following functions. a) \( y=\frac{1}{2 x^{2}}+\sqrt{x} \) b) \( y=\sqrt[3]{x^{2}+5} \) c) \( y=x \sin ^{4} x \) d) \( y=\frac{e^{2 x}}{1+e^{x}} \) e) \( y=\ln1 answer -
1. Simplify \( \frac{(\cos 5 \theta-i \sin 5 \theta)^{2}(\cos 7 \theta+i \sin 7 \theta)^{-3}}{(\cos 8 \theta+i \sin 8 \theta)^{2}(\cos \theta+i \sin \theta)^{5}} \) using De Movier's Theorem.1 answer -
Q1. Simplify \( \frac{(\cos 5 \theta-i \sin 5 \theta)^{2}(\cos 7 \theta+i \sin 7 \theta)^{-3}}{(\cos 8 \theta+i \sin 8 \theta)^{2}(\cos \theta+i \sin \theta)^{5}} \) using De Movier's Theorem.1 answer -
39. Prove each identity. (13 marks) - \( \sin x+\tan x=\tan x(\cos x+1) \) - \( \tan \theta-1=\frac{\sin ^{2} \theta-\cos ^{2} \theta}{\sin \theta \cos \theta+\cos ^{2} \theta} \) - \( \sin x+\sin x \1 answer -
Find \( \int_{0}^{2} f(x, y) d x \) and \( \int_{0}^{3} f(x, y) d y \) \[ f(x, y)=5 x+3 x^{2} y^{2} \] \[ \begin{array}{l} \int_{0}^{2} f(x, y) d x= \\ \int_{0}^{3} f(x, y) d y= \end{array} \]1 answer -
Evaluate the double integral. \[ \iint_{D} \frac{y}{x^{2}+1} d A, D=\{(x, y) \mid 0 \leq x \leq 5,0 \leq y \leq \sqrt{x}\} \]3 answers -
1. Solve the following initial value problems: (a) \( y^{\prime}-\frac{x^{2}}{(y-1)^{2}}=0, \quad y(0)=4 \) (b) \( y^{\prime \prime}+7 y^{\prime}+12 y=4 \sin (3 t), \quad y(0)=\frac{14}{75}, \quad y^{1 answer -
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1. Resolver la ecuación diferencial \[ \frac{d^{2} T(x)}{d x^{2}}=0 \] sujeta a la condición en la frontera \[ \begin{array}{l} T(0)=-T_{0} \\ T(L)=T_{0} \end{array} \] 2. Resolver la ecuación dife1 answer