35- Xdy3dt. where x>9t +22Findfor y =dx9dx

Name: 35- Xdy3dt. where x>9t +22Findfor y =dx9dx
Uploaded: 2024-03-20T07:06:41.000Z
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Description: 35- Xdy3dt. where x>9t +22Findfor y =dx9dx

The given function is y=∫0x39t+2dt. Let u=9t+2. Then, du=9 dt. The upper limit will be 9x+2 and the…

Answered: 3 2 Find for y = dt, where x> 9t+2 dx…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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35-

X
dy
3
dt. where x>
9t +2
2
Find
for y =
dx
9
dx

Expert Solution

Step 1

The given function is $y = \int_{0}^{x} \frac{3}{9 t + 2} d t$ .

Let $u = 9 t + 2$ . Then, $d u = 9 d t$ .

The upper limit will be $9 x + 2$ and the lower limit will be $2$ .

Evaluate the integral as follows.

$\begin{array}{rcl} y & = & \int_{0}^{x} \frac{3}{9 t + 2} d t \\ = & \int_{2}^{9 x + 2} \frac{3}{u} (\frac{1}{9} d u) \\ = & \frac{1}{3} \int_{2}^{9 x + 2} \frac{1}{u} d u \\ = & \frac{1}{3} {[\ln u]}_{2}^{9 x + 2} \\ = & \frac{1}{3} [\ln (9 x + 2) - \ln (2)] \end{array}$

Thus, $y = \begin{array}{rcl} \frac{1}{3} \end{array} \begin{array}{rcl} [\ln (9 x + 2) - \ln (2)] \end{array}$ .

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