solve the initial value problem by separation of variables dy/dx=x2/y given y=-5 when x=3
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To solve the initial value problem dy/dx = x^2/y, we will use the method of separation of variables.
Step 1: Rewrite the equation in separated form.
Start by multiplying both sides of the equation by y to get rid of the denominator:
y * dy/dx = x^2
Step 2: Rearrange the equation to separate the variables.
Now, we rearrange the equation so that all the terms containing y are on one side, and all the terms containing x are on the other side:
y * dy = x^2 * dx
Step 3: Integrate both sides of the equation.
Now we integrate both sides of the equation. Let's integrate the left side first:
∫ y * dy = ∫ x^2 * dx
The integral of y with respect to y is (1/2) * y^2.
The integral of x^2 with respect to x is (1/3) * x^3.
So, the equation becomes:
(1/2) * y^2 = (1/3) * x^3 + C
Note: C is the constant of integration that appears when integrating indefinite integrals.
Step 4: Solve for the constant of integration.
To find the value of the constant of integration (C), we will use the given initial condition y = -5 when x = 3.
Substituting the initial condition into the equation, we have:
(1/2) * (-5)^2 = (1/3) * 3^3 + C
(1/2) * 25 = (1/3) * 27 + C
25/2 = 9/3 + C
25/2 = 3 + C
C = 25/2 - 3
C = 19/2
Step 5: Substitute the value of C back into the equation.
Using the value of C = 19/2, we can rewrite the equation as:
(1/2) * y^2 = (1/3) * x^3 + 19/2
Step 6: Solve for y.
To solve for y, we can multiply both sides of the equation by 2:
y^2 = (2/3) * x^3 + 19
And then take the square root of both sides:
y = ± √((2/3) * x^3 + 19)
Therefore, the solution to the initial value problem dy/dx = x^2/y, with the initial condition y = -5 when x = 3, is:
y = ± √((2/3) * x^3 + 19)