Solve the differential equation:
dy/dx = x (y^(1/3))
x dx=dy/y^1/3
1/2 x^2=y^2/3 * 3/2
x^2=3y^2/3
then put it in any form you wish.
How could I rearrange this to form y=? Out of all the combinations I have tried, I can't find any that satisfy the initial condition f(2)=8.
To solve the given differential equation, we can use the method of variable separation.
Let's assume that y can be expressed as a function of x, y = f(x).
Now, we can rewrite the given differential equation as:
dy/dx = x * (y^(1/3))
Then, we can separate the variables by multiplying both sides by dx and dividing both sides by y^(1/3):
(1/y^(1/3)) * dy = x * dx
Next, we can integrate both sides with respect to their variables. Integrating (1/y^(1/3)) * dy gives us:
∫(1/y^(1/3)) * dy = ∫x * dx
To integrate the left side, we can use the substitution u = y^(1/3), which implies that du = (1/3) * y^(-2/3) * dy. Rearranging, we get dy = 3 * y^(2/3) * du.
Substituting this value of dy in the left side integral, we have:
∫(1/y^(1/3)) * dy = ∫3 * y^(2/3) * du
The integral on the right side is now easier to handle. Integrating both sides, we get:
(3/2) * y^(2/3) = (1/2) * x^2 + C
Here, C is the constant of integration.
Finally, to solve for y, we can multiply both sides by (2/3) and raise both sides to the power of 3/2:
y = [(1/2) * x^2 + C]^(3/2)
Therefore, the solution to the given differential equation is y = [(1/2) * x^2 + C]^(3/2), where C is an arbitrary constant.