What is the solution of the Homogeneous Differential Equation? : #dy/dx = (x^2+y^2-xy)/x^2# with #y(1)=0#
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1 Answer
Explanation:
We have:
# dy/dx = (x^2+y^2-xy)/x^2 # with #y(1)=0#
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:
# v + x(dv)/dx = (x^2+(vx)^2-x(vx))/x^2 #
Then assuming that #x ne 0# this simplifies to:
# v + x(dv)/dx = 1+v^2-v #
Alarme 2 0 100000
# :. x(dv)/dx = v^2-2v+1 #
And we have reduced the initial ODE to a First Order Separable ODE, so we can collect terms and separate the variables to get: