how do you 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, follow these steps:
Step 1: Rearrange the equation.
dy/y^2 = -x^2 dx
Step 2: Separate the variables.
1/y^2 dy = -x^2 dx
Step 3: Integrate both sides.
∫(1/y^2) dy = ∫(-x^2) dx
To integrate 1/y^2 dy, we can use the power rule of integration.
∫(1/y^2) dy = -1/y
To integrate -x^2 dx, we can use the power rule of integration.
∫(-x^2) dx = -x^3/3
Step 4: Apply the initial condition.
Now we plug in the initial condition y(4) = 4 to find the constant of integration.
-1/4 = -4^3/3 + C
Simplifying gives:
-1/4 = -64/3 + C
To solve for C, we can rearrange the equation:
C = -1/4 + 64/3
C = (48 - 256)/12
C = -208/12
C = -52/3
Therefore, the constant of integration C is -52/3.
Step 5: Substitute the constant of integration and solve for y.
-1/y = -x^3/3 - 52/3
Simplifying gives:
1/y = x^3/3 + 52/3
To solve for y, we take the reciprocal of both sides:
y = 3/(x^3/3 + 52/3)
Simplifying further:
y = 9/(x^3 + 52)
So, the solution to the initial value problem dy/dx = -x^2y^2, y(4) = 4 is y = 9/(x^3 + 52).