How can I solve this differential equation? : #xy^2dy/dx=y^3-x^3#
# y = xroot(3)(C - 3ln x) #
We have:
Which is a First Order Nonlinear Ordinary Differential Equation. Let us attempt a substitution of the form:
Substituting into the initial ODE we get:
And we have reduced the initial ODE to a First Order Separable ODE, so we can collect terms and separate the variables to get:
Both integrals are standard, so we can integrate to get:
Then, we restore the substitution, to get the General Solution:
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To solve the differential equation ( xy^2 \frac{dy}{dx} = y^3 - x^3 ), we can use the method of separating variables.
First, divide both sides by ( xy^2 ) to isolate the variables:
[ \frac{dy}{dx} = \frac{y^3 - x^3}{xy^2} ]
Next, we can rewrite the equation as:
[ \frac{1}{y^2} \frac{dy}{dx} = \frac{y - \frac{x^3}{y^2}}{x} ]
Now, we separate the variables by multiplying both sides by ( dx ) and ( y^2 ):
[ y^{-2} dy = \left( y - \frac{x^3}{y^2} \right) dx ]
Integrating both sides yields:
[ \int y^{-2} , dy = \int \left( y - \frac{x^3}{y^2} \right) , dx ]
[ -\frac{1}{y} = \frac{1}{2} y^2 - \frac{x^3}{y^2} + C ]
Multiplying through by ( -y ) to clear the fraction:
[ 1 = -\frac{1}{2} y^3 + x^3 - Cy ]
Rearranging terms:
[ Cy - \frac{1}{2} y^3 = x^3 - 1 ]
This equation represents the general solution to the given differential equation, where ( C ) is the constant of integration.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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