What is the solution of the Homogeneous Differential Equation? : #dy/dx = (x^2+y^2-xy)/x^2# with #y(1)=0#
# y = (xln|x|)/(1+ln|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:
Thus we have:
Then, we restore the substitution, to get the General Solution:
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The solution to the homogeneous differential equation dy/dx = (x^2 + y^2 - xy)/x^2 with the initial condition y(1) = 0 is y(x) = x * (1 - √(2 - x^2)).
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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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