solve the initial value problem by separation of variables dy/dx=-x^2y^2, y(4)=4
To solve the initial value problem dy/dx = -x^2y^2 with the initial condition y(4) = 4 using separation of variables, you can follow these steps:
Step 1: Separate the variables.
∫ y^(-2) dy = ∫ -x^2 dx
Step 2: Evaluate the integrals on both sides.
∫ y^(-2) dy = -∫ x^2 dx
-y^(-1) = -x^3/3 + C
Step 3: Solve for y.
Multiply both sides by -1 to isolate y.
y^(-1) = x^3/3 - C
Now, we need to take the reciprocal on both sides.
y = (1 / (x^3/3 - C))
Step 4: Substitute the initial condition y(4) = 4 to find the value of C.
Plug in the value of x = 4 and y = 4 into the equation.
4 = (1 / (4^3/3 - C))
Solve the equation for C:
4 = (1 / (64/3 - C))
Take the reciprocal on both sides:
1/4 = 64/3 - C
Simplify and solve for C:
C = 64/3 - 1/4
C = (256 - 3) / 12
C = 253/12
Step 5: Substitute the value of C back into the equation.
y = (1 / (x^3/3 - 253/12))
Therefore, the solution to the initial value problem dy/dx = -x^2y^2, y(4) = 4 is y = (1 / (x^3/3 - 253/12)).