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What is the solution of the Homogeneous Differential Equation? : #dy/dx = (x^2+y^2-xy)/x^2# with #y(1)=0#
1 Answer
Explanation:
We have:
Which is a First Order Nonlinear Ordinary Differential Equation. Let us attempt a substitution of the form:
Differentiating wrt #x# and applying the product rule, we get:
Substituting into the initial ODE we get:
![1/2 1/2](https://3.bp.blogspot.com/-iHAHFypDDEI/TcVU03wWrBI/AAAAAAAAABk/juaFT7wdiwk/s1600/Wonderpen-los-564x272.jpg)
Then assuming that #x ne 0# this simplifies to:
And we have reduced the initial ODE to a First Order Separable ODE, so we can collect terms and separate the variables to get:
Little big workshop mac. Both integrals are standard, so we can integrate to get:
Using the initial condition, # y(1)=0 => v(1)=0 # , we get:
Thus we have:
Typinator 7 9 day. # -1/(v-1) = ln|x| +1 #
Then, we restore the substitution, to get the General Solution: