solve the value problem by separation of variables dy/dx=x^2/y when y=-5 and x=3
To solve the value problem dy/dx = x^2/y using separation of variables, we must separate the variables and then integrate both sides. Here's how you can do it step by step:
Step 1: Rewrite the equation in a separated form:
yy' = x^2
Step 2: Separate the variables by putting all terms involving y on one side and all terms involving x on the other side:
y dy = x^2 dx
Step 3: Integrate both sides of the equation with respect to their respective variables:
∫ y dy = ∫ x^2 dx
Step 4: Solve the definite integral on each side:
(1/2) y^2 = (1/3) x^3 + C
Where C is the constant of integration.
Step 5: Apply the initial condition y = -5 when x = 3 to find the value of the constant C:
(1/2) (-5)^2 = (1/3) (3^3) + C
25/2 = 27/3 + C
25/2 = 9 + C
C = 25/2 - 9 = 25/2 - 18/2 = 7/2
Step 6: Substitute the value of C back into the general solution:
(1/2) y^2 = (1/3) x^3 + 7/2
So, the solution to the value problem dy/dx = x^2/y when y = -5 and x = 3 is:
(1/2) y^2 = (1/3) x^3 + 7/2