solve the initial value problem by separation of variables dy/dx=-x^2y^2, y(4)=4
These are all similar to your previous problem (under Billy):
http://www.jiskha.com/display.cgi?id=1300573436
To solve the given initial value problem dy/dx = -x^2y^2 with the initial condition y(4) = 4, we can use the method of separation of variables.
Step 1: Separate the variables
Start by separating the variables, which means putting all the x terms on one side of the equation and all the y terms on the other side.
Divide both sides of the equation by y^2:
dy/y^2 = -x^2 dx
Step 2: Integrate both sides
Next, integrate both sides of the equation with respect to their respective variables.
∫(dy/y^2) = ∫(-x^2 dx)
The integral of dy/y^2 can be found by using substitution. Let u = y, then du = dy.
∫(1/y^2) du = ∫(-x^2 dx)
Integrating both sides, we get:
(-1/y) = (-1/3) x^3 + C
where C is the constant of integration.
Step 3: Solve for y
Now, solve the equation for y by isolating y on one side. Multiply both sides by -1:
1/y = (1/3) x^3 - C
To get rid of the fraction, take the reciprocal of both sides:
y = 1 / ((1/3) x^3 - C)
Step 4: Apply the initial condition
Now, substitute the initial condition y(4) = 4 into the equation we obtained in step 3 and solve for C.
4 = 1 / ((1/3) (4)^3 - C)
Simplify the expression:
4 = 1 / ((64/3) - C)
Multiply both sides by ((64/3) - C):
4((64/3) - C) = 1
Simplify further:
(256/3) - 4C = 1
Now, isolate C:
-4C = 1 - (256/3)
-4C = -253/3
C = (253/12)
Step 5: Final solution
Finally, substitute the value of C back into the equation in step 3:
y = 1 / ((1/3) x^3 - (253/12))
Therefore, the solution to the initial value problem dy/dx = -x^2y^2 with the initial condition y(4) = 4 is:
y = 1 / ((1/3) x^3 - (253/12))