Differentiation Difficulties! Partial Differentiation.?
Every time I try and work this out, I keep getting 0 overall on the left side. I think I'm doing it totally wrong
f(x; y) = e^x sin(2y):
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d^2f/dx^2 - d^2f/dy^2 = 5f:
d^2f/dx^2 of e^x sin(2y) is 0 right? There's no x^2 so everything's constant and thus goes to 0 through differentiation. Is this wrong!?
Answers & Comments
Verified answer
f(x,y) = e^x sin(2y)
∂f/∂x = e^x sin(2y) -----> sin(2y) is treated as constant
∂²f/∂x² = e^x sin(2y)
∂f/∂y = 2 e^x cos(2y) -----> e^x is treated as constant
∂²f/∂x² = -4 e^x sin(2y)
∂²f/∂x² - ∂²f/∂x²
= e^x sin(2y) + 4 e^x sin(2y)
= 5 e^x sin(2y)
= 5 f(x,y)
Mαthmφm
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F = e^x sin(2y)
Fx = differentiate F w.r.t.x ...treating y term as CONSTANT
Fx = e^x sin(2y)
Fxx = differentiate Fx w.r.t.x ...treating y term as CONSTANT = d^2f/dx^2
Fxx = e^x sin(2y)
--------------
Fy= differentiate F w.r.t.y ...treating x term as CONSTANT
Fy = e^x * cos(2y) * 2
=2e^x * cos(2y)
Fyy =differentiate Fy w.r.t.y ...treating x term as CONSTANT
Fyy = 2e^x * (-sin(2y)) * 2
= -4e^x * sin(2y)
therefore ....
Fxx --Fyy = e^x sin(2y) -- [-4e^x * sin(2y)]
= 5e^x sin(2y)
= 5F