Using clairaut's form find general solution of p=log(px-y)
Answers
Given:
The general equation p = log(px - y)
To find:
Using clairaut's form find the general solution of p = log(px - y)
Solution:
From given, we have an equation,
p = log(px - y)
⇒ log_e (px - y) = p
⇒ px - y = e^p
∴ y = px - e^p ............(1)
differentiating both sides wrt x, we get,
dy/dx = d/dx (px - e^p)
dy/dx = p + x dp/dx - e^p dp/dx
∴ p = p + (x - e^p) dp/dx (∵ dy/dx = p)
(x - e^p) dp/dx = 0 ...........(2)
(x - e^p) ≠ 0, when dp/dx = 0
∴ dp/dx = 0
dp = 0 dx
integrating on both sides, we get,
∫ dp = ∫ 0 dx
p = c ( c = constant) ...........(3)
from (1) and (3), we get,
y = cx - e^c [ this is the general solution ]
from (2),
if x - e^p = 0
⇒ x = e^p
log x = p
∴ p = log x ..........(4)
from (1) and (4), we get,
y = log x × x - e^{log x}
y = x log x - x
y = x (log x - 1) [ this is the singular solution ]
Answer:
CHECK IT OUT
Step-by-step explanation:
h t t p s : / / w w w .y o u t u b e . c o m / w a t c h ? v = E b x H S Z _ v x 2 g