How can I solve this differential equation? : #xy^2dy/dx=y^3-x^3#

Answer 1

# y = xroot(3)(C - 3ln x) #

We have:

# xy^2dy/dx=y^3-x^3 #

Which is a First Order Nonlinear Ordinary Differential Equation. Let us attempt a substitution of the form:

# y = vx #
Differentiating wrt #x# and applying the product rule, we get:
# dy/dx = v + x(dv)/dx #

Substituting into the initial ODE we get:

# x(vx)^2(v + x(dv)/dx) = (vx)^3 - x^3 #
Then assuming that #v,x ne 0# this simplifies to:
# v^2(v + x(dv)/dx) = v^3 -1 #
# :. v + x(dv)/dx = v -1/v^2 #
# :. x(dv)/dx = -1/v^2 #

And we have reduced the initial ODE to a First Order Separable ODE, so we can collect terms and separate the variables to get:

# int \ v^2 \ dv = int \ -1/x \ dx #

Both integrals are standard, so we can integrate to get:

# 1/3 v^3 = - ln x + A #
# :. v^3 = 3A - 3ln x #
# :. v = root(3)(C - 3ln x) \ \ \ #, say

Then, we restore the substitution, to get the General Solution:

# y/x = root(3)(C - 3ln x) #
# :. y = xroot(3)(C - 3ln x) #
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Answer 2

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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Answer from HIX Tutor

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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