how do you solve the variable dy/dx=-x^2y^2, y(4)=4
To solve the given differential equation dy/dx = -x^2y^2 with the initial condition y(4) = 4, we can use the method of separation of variables.
Step 1: Write the differential equation in the following form:
dy/y^2 = -x^2 dx
Step 2: Integrate both sides of the equation with respect to their respective variables.
∫ dy/y^2 = - ∫ x^2 dx
Step 3: Evaluate the integrals.
The integral of dy/y^2 is (-1/y).
The integral of x^2 dx is (x^3)/3, by using the power rule of integration.
Therefore, the equation becomes:
-1/y = (-x^3)/3 + C (where C is the constant of integration)
Step 4: Apply the initial condition y(4) = 4 to find the value of C.
-1/4 = (-4^3)/3 + C
Simplify the equation:
-1/4 = -64/3 + C
Find the value of C by solving the equation:
C = -1/4 + 64/3
C = -1/4 + 256/12
C = -1/4 + 64/12
C = -3/12 + 64/12
C = 61/12
Step 5: Substitute the value of C back into the general solution:
-1/y = (-x^3)/3 + 61/12
Step 6: Solve for y by isolating it:
1/y = (x^3)/3 - 61/12
Take the reciprocal of both sides:
y = 1 / [(x^3)/3 - 61/12]
This is the general solution to the given differential equation.