To find the value of the expression, we first simplify the given differential equation.
Step 1: Simplify the differential equation
The given equation is ex+ydxdy=ex−y. Using exponent laws (ex+y=ex⋅ey and ex−y=ex⋅e−y), we get:
(ex⋅ey)dxdy=ex⋅e−y
Dividing both sides by ex (since ex=0):
eydxdy=e−y
e2ydy=dx
Step 2: Find the derivatives
From the simplified form dx=e2ydy, we have:
dydx=e2y
Taking the reciprocal, we get:
dxdy=e2y1=e−2y
Now, differentiate dxdy with respect to x to find dx2d2y:
dx2d2y=dxd(e−2y)=−2e−2ydxdy
Substitute dxdy=e−2y into this:
dx2d2y=−2e−2y(e−2y)=−2e−4y
Step 3: Evaluate the expression
We want to find dx2d2y(dydx)2:
Expression=(−2e−4y)⋅(e2y)2
Expression=(−2e−4y)⋅(e4y)
Expression=−2e0=−2
Therefore, the value is −2, which corresponds to option (a).
