solve the differential equations subject to the given conditions.
dy/dx= 4x^1/2 y=21 of x=4
is this derivative?!
**antidervative
dy=4x^.5 dx
y= 2/3 * 4 x^1.5 + C
21=8/3 * 4^3/2 + C
21=64/3+C solve for C, then you have it.
Yes, "dy/dx" represents the derivative of y with respect to x. In this case, the equation is a first-order ordinary differential equation. And the conditions given are "y=21" when "x=4".
Yes, the equation "dy/dx = 4x^(1/2)y" represents a differential equation. It is a first-order ordinary differential equation because it involves the first derivative of an unknown function y(x) with respect to x.
To solve this differential equation, we need to find the function y(x) that satisfies the equation and the given conditions.
The given condition is y = 21 when x = 4.
To solve this differential equation, we can use the method of separation of variables. Here's how:
Step 1: Rewrite the equation in separated form.
dy/y = 4x^(1/2) dx
Step 2: Integrate both sides of the equation.
∫ (dy/y) = ∫ (4x^(1/2) dx)
Integrating the left side gives:
ln|y| = ∫ (4x^(1/2) dx)
Integrating the right side gives:
ln|y| = (8/3) * x^(3/2) + C, where C is the integration constant.
Step 3: Apply the initial condition to determine the value of the integration constant.
Using the given condition y = 21 when x = 4:
ln|21| = (8/3) * (4)^(3/2) + C
ln|21| = (8/3) * 8 + C
ln|21| = 64/3 + C
Step 4: Solve for C.
C = ln|21| - 64/3
Therefore, the solution to the differential equation with the given condition is:
ln|y| = (8/3) * x^(3/2) + ln|21| - 64/3
To get the actual function y(x), we need to take the exponential of both sides:
|y| = e^((8/3) * x^(3/2) + ln|21| - 64/3)
Since the absolute value signs can be dropped due to the initial condition y = 21 when x = 4, we can have:
y = e^((8/3) * x^(3/2) + ln|21| - 64/3)
This is the solution to the given differential equation subject to the condition y = 21 when x = 4.