Math  /  Calculus

Questiona. Find zttz_{t t} for ztt=z(x,y,t)=cos(4+49t)sin(2x)sin(7y)z_{t t}=\quad z(x, y, t)=\cos (\sqrt{4+49 t}) \cdot \sin (2 x) \cdot \sin (7 y) aba^{b} sin(a)xf\sin (a) \quad \frac{\partial}{\partial x} f : \infty α\alpha Ω\Omega ? b. Does u=sin(53t)sin(2x)sin(7y)u=\sin (\sqrt{53} \cdot t) \cdot \sin (2 x) \cdot \sin (7 y) satisfy the membrane equation utt=uxx+uyy?u_{t t}=u_{x x}+u_{y y} ? Yes No

Studdy Solution

STEP 1

What is this asking? We need to find the second partial derivative of zz with respect to tt, and then check if a different function uu satisfies the membrane equation, which relates its second partial derivatives with respect to time, xx, and yy. Watch out! Don't mix up xx, yy, and tt when taking derivatives, and be careful with the chain rule!
Also, remember the membrane equation needs the sum of the second spatial derivatives, not the difference.

STEP 2

1. Calculate ztt
2. Check the membrane equation

STEP 3

Alright, let's **dive in** and find zttz_{tt}!
We're given z(x,y,t)=cos(4+49t)sin(2x)sin(7y)z(x, y, t) = \cos (\sqrt{4 + 49t}) \cdot \sin(2x) \cdot \sin(7y).
Since we're taking the derivative with respect to tt, we can treat sin(2x)\sin(2x) and sin(7y)\sin(7y) as **constants**.

STEP 4

First, let's find ztz_t, the **first derivative** of zz with respect to tt.
Using the chain rule, we have: zt=sin(4+49t)ddt(4+49t)sin(2x)sin(7y) z_t = -\sin(\sqrt{4 + 49t}) \cdot \frac{d}{dt}(\sqrt{4 + 49t}) \cdot \sin(2x) \cdot \sin(7y)

STEP 5

Now, let's find ddt(4+49t)\frac{d}{dt}(\sqrt{4 + 49t}).
Remember, 4+49t\sqrt{4 + 49t} is the same as (4+49t)12(4 + 49t)^{\frac{1}{2}}.
Using the chain rule again: ddt(4+49t)12=12(4+49t)1249=4924+49t \frac{d}{dt}(4 + 49t)^{\frac{1}{2}} = \frac{1}{2}(4 + 49t)^{-\frac{1}{2}} \cdot 49 = \frac{49}{2\sqrt{4 + 49t}}

STEP 6

Substituting this back into our expression for ztz_t: zt=sin(4+49t)4924+49tsin(2x)sin(7y) z_t = -\sin(\sqrt{4 + 49t}) \cdot \frac{49}{2\sqrt{4 + 49t}} \cdot \sin(2x) \cdot \sin(7y)

STEP 7

Now, for the **grand finale**, let's find zttz_{tt}.
We'll need the product rule and the chain rule!
Deep breaths, we've got this! ztt=[cos(4+49t)4924+49t4924+49tsin(4+49t)(494(4+49t)3249)]sin(2x)sin(7y) z_{tt} = \left[ -\cos(\sqrt{4 + 49t}) \cdot \frac{49}{2\sqrt{4 + 49t}} \cdot \frac{49}{2\sqrt{4 + 49t}} - \sin(\sqrt{4 + 49t}) \cdot \left( -\frac{49}{4} (4 + 49t)^{-\frac{3}{2}} \cdot 49 \right) \right] \sin(2x) \sin(7y)

STEP 8

Simplifying zttz_{tt}: ztt=[cos(4+49t)24014(4+49t)+sin(4+49t)24014(4+49t)32]sin(2x)sin(7y) z_{tt} = \left[ -\cos(\sqrt{4 + 49t}) \cdot \frac{2401}{4(4 + 49t)} + \sin(\sqrt{4 + 49t}) \cdot \frac{2401}{4(4 + 49t)^{\frac{3}{2}}} \right] \sin(2x) \sin(7y)

STEP 9

Now, let's see if u=sin(53t)sin(2x)sin(7y)u = \sin(\sqrt{53}t) \cdot \sin(2x) \cdot \sin(7y) satisfies the membrane equation utt=uxx+uyyu_{tt} = u_{xx} + u_{yy}.

STEP 10

First, let's find uttu_{tt}.
Remember, treat sin(2x)\sin(2x) and sin(7y)\sin(7y) as **constants**. ut=53cos(53t)sin(2x)sin(7y) u_t = \sqrt{53}\cos(\sqrt{53}t) \cdot \sin(2x) \cdot \sin(7y) utt=53sin(53t)sin(2x)sin(7y) u_{tt} = -53\sin(\sqrt{53}t) \cdot \sin(2x) \cdot \sin(7y)

STEP 11

Next, let's find uxxu_{xx}.
This time, sin(53t)\sin(\sqrt{53}t) and sin(7y)\sin(7y) are our **constant friends**. ux=sin(53t)2cos(2x)sin(7y) u_x = \sin(\sqrt{53}t) \cdot 2\cos(2x) \cdot \sin(7y) uxx=sin(53t)(4sin(2x))sin(7y)=4sin(53t)sin(2x)sin(7y) u_{xx} = \sin(\sqrt{53}t) \cdot (-4\sin(2x)) \cdot \sin(7y) = -4\sin(\sqrt{53}t) \sin(2x) \sin(7y)

STEP 12

Finally, let's calculate uyyu_{yy}.
Now, sin(53t)\sin(\sqrt{53}t) and sin(2x)\sin(2x) are **constants**. uy=sin(53t)sin(2x)7cos(7y) u_y = \sin(\sqrt{53}t) \cdot \sin(2x) \cdot 7\cos(7y) uyy=sin(53t)sin(2x)(49sin(7y))=49sin(53t)sin(2x)sin(7y) u_{yy} = \sin(\sqrt{53}t) \cdot \sin(2x) \cdot (-49\sin(7y)) = -49\sin(\sqrt{53}t) \sin(2x) \sin(7y)

STEP 13

Let's check if utt=uxx+uyyu_{tt} = u_{xx} + u_{yy}: 53sin(53t)sin(2x)sin(7y)=4sin(53t)sin(2x)sin(7y)49sin(53t)sin(2x)sin(7y) -53\sin(\sqrt{53}t)\sin(2x)\sin(7y) = -4\sin(\sqrt{53}t)\sin(2x)\sin(7y) - 49\sin(\sqrt{53}t)\sin(2x)\sin(7y) 53sin(53t)sin(2x)sin(7y)=53sin(53t)sin(2x)sin(7y) -53\sin(\sqrt{53}t)\sin(2x)\sin(7y) = -53\sin(\sqrt{53}t)\sin(2x)\sin(7y) It **matches**!

STEP 14

ztt=[cos(4+49t)24014(4+49t)+sin(4+49t)24014(4+49t)32]sin(2x)sin(7y)z_{tt} = \left[ -\cos(\sqrt{4 + 49t}) \cdot \frac{2401}{4(4 + 49t)} + \sin(\sqrt{4 + 49t}) \cdot \frac{2401}{4(4 + 49t)^{\frac{3}{2}}} \right] \sin(2x) \sin(7y).
The function uu *does* satisfy the membrane equation.

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