Which of the following is the general solution of the differential equation dy dx equals the quotient of 8 times x and y?
do you mean product of x and y?
dy/dx = 8 x y
or do you mean
dy/dx = 8 x / y
or what ?
if the latter then
y dy = 8 x dx
y^2/2 = 8 x^2/2 + C
y^2 - 8 x^2 = constant
y'=8x/y
separation of variables...
y dy = 8x dx
1/2 y^2 + C=8/2x^2 + K
looks to me the general solution is
y^2=8 x^2 + K
check:
2ydy=16x dx
dy/dx=8x/y
We agree, if that is indeed the question :)
To find the general solution of the given differential equation, we need to separate variables and then integrate.
Step 1: Rewrite the given differential equation in the appropriate form.
The differential equation is given as:
dy/dx = 8x/y
Step 2: Cross-multiply variables to separate them.
Multiplying both sides by y:
ydy = 8xdx
Step 3: Integrate both sides of the equation.
Integrating the left side with respect to y and the right side with respect to x:
∫ydy = ∫8xdx
Step 4: Perform the integrations.
Integrating the left side gives:
(1/2)y^2
Integrating the right side gives:
4x^2 + C (where C is the constant of integration)
Therefore, the general solution of the given differential equation is:
(1/2)y^2 = 4x^2 + C
Note: The constant of integration (C) represents an arbitrary constant that accounts for the possible solutions missed during integration.