Final answer:
In order to find the value of (d^(2)y)/(dx^(2)) at the point (3,0), differentiate the given expression (x+2y)*(dy)/(dx)=2x-y twice with respect to x. Substituting the point (3,0) and solving the equation, you can find the value of (d^(2)y)/(dx^(2)).
Step-by-step explanation:
To find the value of (d^(2)y)/(dx^(2)) at the point (3,0), we need to differentiate the given expression (x+2y)*(dy)/(dx)=2x-y twice with respect to x.
Let's start by differentiating the expression once: (x+2y)*(dy)/(dx)=2x-y
Using the product rule, we have: (dy)/(dx) + (x+2y)*((d^2y)/(dx^2)) = 2 - (dy)/(dx)
Now, differentiate again with respect to x: (d^2y)/(dx^2) + ((dy)/(dx))*((dy)/(dx)) + (x+2y)*((d^2y)/(dx^2)) = 0
Substituting the point (3,0), we have: (d^2y)/(dx^2) + ((dy)/(dx))*((dy)/(dx)) + (3+2*0)*((d^2y)/(dx^2)) = 0
Simplifying the equation, we get: (d^2y)/(dx^2) + ((dy)/(dx))^2 + 3*((d^2y)/(dx^2)) = 0
Combining like terms, we have:
4*((d^2y)/(dx^2)) + ((dy)/(dx))^2 = 0
Substituting the value of x as 3, we get:
4*((d^2y)/(dx^2)) + ((dy)/(dx))^2 = 0
Now, we need to solve this equation to find the value of (d^(2)y)/(dx^(2)) at the point (3,0).