I don’t understand how to solve this one: y’=(2x-2y+3)/(3x-7y+1). I need to make a replacement, but I don’t know what I should take for u(x). What is more, I’m not sure if this differential equation is homogeneous at all. When I add λ, it turns into something like this: y’=(2xλ-2yλ+3)/(3λx-7yλ+1) and I can’t divide it so that the equation will be the same.
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It's not homogeneous, so first you need a substitution that makes it homogeneous. – Gerry Myerson Nov 14 '20 at 11:21
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I decide to make a substitution like this: u(x)=(y(x)+7/8)/(x+19/8). But I still have problems with λ, it doesn’t go away. – Katya5333 Nov 14 '20 at 11:38
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I don't know what you mean by $\lambda$, but $x=u-(19/8)$, $y=v-(7/8)$ should transform your equation to $dv/du=(2u-2v)/(3u-7v)$, which is homogeneous. Now letting $v=uz$ should lead to a variables-separable equation for $dz/du$. – Gerry Myerson Nov 14 '20 at 23:08
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Did that work for you? – Gerry Myerson Nov 16 '20 at 12:14