Advanced Math Archive: Questions from October 31, 2022
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Figure 8.13.1 Eigenfunctions \( \psi_{n} \) for \( n=0,1,2,3 \). 8.14 Exercises 1. Determine the eigenvalues and eigenfunctions of the following regular Sturm-Liouville systems: (a) \( y^{\prime \prim2 answers -
4. Find the eigenvalues and eigenfunctions of the following regular SturmLiouville systems: (a) \( x^{2} y^{\prime \prime}+3 x y^{\prime}+\lambda y=0,1 \leq x \leq e \) \( y(1)=0, y(e)=0 \) (b) \( \fr2 answers -
2 answers
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Solve the differential equation y' - 9y = 18
Resuelva la ecuación diferencial \( \mathrm{y}^{\prime}-9 \mathrm{y}=18 \)2 answers -
Solve the differential equation (2y - 2)y' = 2x - 1 If in the previous exercise y(1) = 1, find a particular solution of the differential equation
Resuelva la ecuación diferencial \( (2 \mathrm{y}-2) \mathrm{y}^{\prime}=2 \mathrm{x}-1 \) Si en el ejercicio anterior \( y(1)=1 \), encuentre una solución particular de la ecuación diferencial2 answers -
Solve the differential equation dy/dx + 2y = g(x), y(0) = - 1, where g(x) = { 0 if x > 3 1 if 0≤x≤3
Resuelva la ecuación diferencial \( \frac{d y}{d x}+2 y=g(x), y(0)=-1 \), donde \( g(x)=\left\{\begin{array}{ll}0 & \text { si } x>3\end{array}\right. \) 1 si \( 0 \leq x \leq 3 \)2 answers -
Determine whether the following statement is true or false: x^2y+ y^2 = 1 is a solution of the exact equation 2xydx + (×^2 + y)dy = 0
Determine si la siguiente afirmación es verdadera o falsa: \( \mathrm{x}^{2} \mathrm{y}+\mathrm{y}^{2}=1 \) es solución de la ecuación exacta \( 2 \mathrm{xydx}+\left(\mathrm{x}^{2}+\mathrm{y}\righ2 answers -
Solve the initial value problem \[ \mathbf{y}^{\prime}=\left[\begin{array}{ccc} -1 & 4 & 2 \\ -2 & 5 & 2 \\ 1 & -2 & 0 \end{array}\right] \mathbf{y}, \quad \mathbf{y}(0)=\left[\begin{array}{c} 7 \\ 52 answers -
\[ 9 y x^{2}-6 x(y+1)+y+1=0 \] Solve for \( \mathrm{x} \) when \( 2 \frac{d^{2} y}{d x^{2}}=6 x+1 \)2 answers -
2 answers
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(11) The genered Soldion for tha DE: \( y^{\prime}=3 x^{2}-2 x+4 \) A) \( y=3 x^{3}-2 x^{2}+4 x+c \) B) \( y=x^{3}-x^{2}+4 x+c \) c) \( y=\frac{3}{4} x^{3}-\frac{2}{3} x^{2}+4 x+c \) D) \( y=\frac{2}{2 answers -
solve differential equations
(1) \( x \frac{d y}{d x}-y=\frac{x^{3}}{y} e^{y / x} \). (2) \( \left(4 x^{2}+2 x y+y\right) d x+\left(x^{2}+y^{3}+x\right) d y=0 \)2 answers -
0. Solve: \( y^{\prime \prime}+y=x^{2}-2 x+e^{3 x}-\sin (4 x) \) 1. Solve: \( y^{\prime \prime}-y=\mathrm{e}^{x^{2}} \) with \( y(0)=1 \) and \( y^{\prime}(0)=5 \).2 answers -
12. Solve : \( x y^{\prime \prime}-y^{\prime}=x^{2} \ln (x) \). 13. Solve: \( y^{\prime \prime}-y=6 \mathrm{e}^{2 x} \) with \( y(0)=5 \) and \( y^{\prime}(0)=2 \)2 answers -
In problems 11-18, relate the set of points given in the figure to the domain of one of the functions in a)-h).
En los problemas 11-18, relacione el conjunto de puntos dados en la figura con el dominio de una de las funciones en \( a \) )- \( h \) ). a) \( f(x, y)=\sqrt{y-x^{2}} \) b) \( f(x, y)=\ln \left(x-y^{2 answers -
\( \lim _{x \rightarrow x_{0}} f(x)=L>0 \), then \( \lim _{x \rightarrow x_{0}} \sqrt{f(x)}=\sqrt{L} \)2 answers -
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14) If \( 3^{x}=a \), evaluate \( \log _{3} a^{4} \) 15) Given \( \frac{y^{2}}{x z^{3}}=3 \), evaluate \( \log _{3}\left(\frac{1}{x}\right)+2 \log _{3} y-\log _{3} z^{3} \)2 answers -
prove v13, v14 , v13=v14
\( V_{13}=(-x)^{-\beta_{2}} F_{1}\left(\beta+1-\gamma, \beta ; \beta+1-\alpha ; \frac{1}{x}\right) \) \( V_{14}=(-x)^{x-\gamma}(1-x)^{\gamma-\alpha-\beta_{2}} F_{1}\left(1-\alpha, \gamma-\alpha ; \bet0 answers -
Sketch the graphs of \( y=f(x) \) and \( y=g(x) \). \[ f(x)=x^{3}-1 ; g(x)=10 x+11 \] Determine all the points of intersecton.2 answers -
28. \( 6 \times 10^{9}+2 \times 10^{-3} \) 30. \( 2 \times 10^{4}+6 \times 10^{2}+4 \times 10^{-1}+7 \times 10^{-3} \) \( +6 \times 10^{-4}+9 \times 10^{-5} \)2 answers -
Solve one of the following ODEs analytically: (a) \( \cos (x) \cdot \frac{d y}{d x}=y \cdot \sin (x)-2 x \) \( y(0)=1 \) (b) \( y^{\prime \prime \prime}+7 \cdot y^{\prime \prime}+16 \cdot y^{\prime}+12 answers -
Solve the given initial value problem \[ \begin{array}{l} y^{\prime \prime \prime}-y^{\prime \prime}-25 y^{\prime}+25 y=0 \\ y(0)=3, \quad y^{\prime}(0)=85, \quad y^{\prime \prime}(0)=-45 \end{array}2 answers -
Sea \( E=\left\{(x, y, z): 4 \leq x^{2}+y^{2} \leq 9,0 \leq z \leq 5-x-y\right\} \) y calculemos la integral \( \iiint_{E} x d V \). Se puede calcular la integral triple en coordenadas cartesianas o e2 answers -
In Problems 1-3, solve the initial value problem: 1. \( y^{\prime \prime}+y^{\prime}-6 y=0, y(0)=1, y^{\prime}(0)=-1 \). 2. \( y^{\prime \prime}-4 y^{\prime}+4 y=0, y(0)=2, y^{\prime}(0)=1 \). 3. \( y2 answers -
2 answers
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Solve the given initial value problem. y''' - 4y'' + 16y' - 24y = 0. y(0) = 1, y'(0) = 0, y''(0) = 0.
Solve the given initial value problem. \[ \begin{array}{l} y^{\prime \prime \prime}-4 y^{\prime \prime}+16 y^{\prime}-24 y=0 \\ y(0)=1, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=0 \end{array}2 answers -
1 answer
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4. \( h(t)=\sec ^{-1}(t)-t^{3} \cos ^{-1}(t) \) 5. \( f(w)=\left(w-w^{2}\right) \sin ^{-1}(w) \) 6. \( y=\left(x-\cot ^{-1}(x)\right)\left(1+\csc ^{-1}(x)\right) \) 7. \( Q(z)=\frac{z+1}{\tan ^{-1}(z)0 answers -
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Se define el diferencial de arco como \( d s=\left\|r^{\prime}(t)\right\| d t \) y la longitud de una curva \( C \) como \( \int_{C} d s=\int_{a}^{b}\left\|r^{\prime}(t)\right\| d t \). Utilice una ap0 answers -
3. Compute the following integrals: (a) Let \( D:=\left\{(x, y): x \geq 0, x^{2}+y^{2} \leq 16\right\} \). Find \[ \iint_{D} x d A \text {. } \] (b) Let \( D:=\left\{(x, y): y \geq 0,1 \leq x^{2}+y^{22 answers -
4. Differentiate (Do not simplify). \( (12, \mathrm{~K}] \) a) \( y=\frac{(x+4)^{2}}{(x-6)^{6}} \) b) \( y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \) c) \( y=300(7)^{(3 x-1)} \) d) \( y=(x+2)^{2} 4^{x} \)2 answers -
all parts
I. Find all the local maxima, local minima, and saddle points of the functions. 1. \( f(x, y)=x^{2}+x y+y^{2}+3 x-3 y+4 \) 2. \( f(x, y)=x^{2}+x y+3 x+2 y+5 \) 3. \( f(x, y)=x^{2}-4 x y+y^{2}+6 y+2 \)2 answers -
1 answer