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What is the solution of the Differential Equation #dy/dx=(x-3)y^2/x^3#?

Answer 1

# y = (2x^2)/(2x-3+Ax^2)#

We have:

#dy/dx=(x-3)y^2/x^3#

This is a first Order non-linear Separable Differential Equation, we can collect terms by rearranging the equation as follows

# 1/y^2 dy/dx=(x-3)/x^3 #

And now we can "separate the variables" to get

# int \ 1/y^2 \ dy= int \ (x-3)/x^3 \ dx #

# :. int \ 1/y^2 \ dy= int \ 1/x^2-3/x^3 \ dx #

And integrating gives us:

# y^(-1)/(-1) = x^(-1)/(-1)-3x^(-2)/(-2) + C_1#

# :. -1/y = -1/x+3/(2x^2) + C_1#

# :. -1/y = (3+2C_1x^2-2x)/(2x^2)#

# :. 1/y = (2x-3+Ax^2)/(2x^2)#

# y = (2x^2)/(2x-3+Ax^2)#

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Answer 2

Isaiah Allen

#-1/y=-1/x+3/(2x^2)+C#

Use separation of variables, that is put the #y# terms in the left, and #x# terms in the right.

#dy/(dx)=(x-3)y^2/x^3#

#(1/y^2)dy/(dx)=(x-3)/x^3#

#(1/y^2)dy=(x-3)/x^3dx#

Integrate both sides:

#int (1/y^2)dy=int (x-3)/x^3dx#

#-1/y=int1/x^2-3/x^3dx#

#-1/y=-1/x-(-3/(2x^2))+C#

#-1/y=-1/x+3/(2x^2)+C#

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Answer 3

William Abdalla

The solution to the differential equation ( \frac{dy}{dx} = \frac{(x-3)y^2}{x^3} ) is ( y = \frac{1}{c - \frac{1}{x^2}} ), where ( c ) is an arbitrary constant.

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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.