The expression dy/dx = x(cbrt(y)) gives the slope at any point on the graph of the function f(x) where f(2) = 8. B. Write an expression for f(x) in terms of x.
I can't figure out what to do with that cubed root; it keeps throwing me off.
dy/dx = x y^(1/3)
y^(-1/3) dy = x dx
(3/2)y^(2/3) = (1/2)x^2 + c
when x = 2 , y = 8
(3/2)(4) = (1/2)(4) + c
c = 6 - 2 = 4
(3/2) y^(2/3) = (1/2) x^2 + 4
3 y^(2/3) = x^2 + 8
y^(2/3) = (1/3) x^2 + 8/3
And then you could simplify it out to get y = (x^2/3 + 8/3)^(3/2). Thank you so much, I went back and ran (2, 8) through and it finally worked out, thanks!!!
your wrong
Strange, I did not write that.
All I did was solve for y so I could get the y by itself. I need to right the expression in terms of x, so I need to isolate y completely. In order to do this, I just applied the 2/3 root to both sides of the equation. Put in different terms, I could also say that it is to the 3/2 power. I worked it through with the initial condition f(2)=8, and it worked out. So, would you put your stamp of approval on this simplification.
Yes, that is correct. One answer you could receive is y= (1/3)^(3/2) * (x^2 + 8)^(3/2). The 1/3 in the beginning is just taking the dividing of 3 from the other components of the equation just to make it simpler.
To find an expression for f(x) in terms of x, we can solve the given differential equation by separating variables.
Starting with the differential equation dy/dx = x(cbrt(y)), we can rewrite it as:
cbrt(y) dy = x dx
Now, we integrate both sides of the equation separately.
Integrating the left-hand side involves integrating cbrt(y) dy. To do this, we can substitute u = y^(1/3), and hence du = (1/3)y^(-2/3) dy:
∫ cbrt(y) dy = ∫ u du = (1/2)u^2 + C1
For the right-hand side, we integrate x dx:
∫ x dx = (1/2)x^2 + C2
Combining these results, we get:
(1/2)y^(1/3) + C1 = (1/2)x^2 + C2
Now, using the given initial condition, f(2) = 8, we can find the values of the constants C1 and C2.
Replacing x with 2 and y with 8 in the equation, we have:
(1/2)8^(1/3) + C1 = (1/2)2^2 + C2
Simplifying, we get:
4^(1/3) + C1 = 2 + C2
Taking the cube root of 4, we find that 4^(1/3) = 2. Substitute this into the equation:
2 + C1 = 2 + C2
From here, we can see that C1 = C2.
Therefore, the expression for f(x) in terms of x is:
(1/2)y^(1/3) + C1 = (1/2)x^2 + C1