Question

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Delermine which of the following diffevential euahons are Scparable. In eaon case exphin why the diffeerhial eaueton is or is not seporable If separable, find the genesal soluhion to tme oiff arential equomon b) y- In(2) in (y)

Answer 1

Given that dy/dx = n (x - y) rightarrow (1) the differential equation is not separable because it cannot be written in the form f(X) dx = g(y) dy but it can be reduced to the form which the variable are separable as follows put x - y = v 1 - dy/dx = dv/dx so dy/dx = 1 - dv/dx substituting in (1) we get 1 - dv/dx = ln(v) this is variable separable and it is solved as follows. Dv/dx = 1- ln v dv/1 - ln v = dx integrating both sides we get integration dv/1 = ln (v) = x + c here the solution on LHS cannot be integral in terms of elementary functions. IT is left in quadrature in terms of integral given that y^derivative = ln(X)/ln (y) A first order separable ODE has the forms of N(y)y^derivative = mu(x) now rewrite in the form of first order separable ODE. Ln(y) y^derivative = ln (x) N(y). y^derivative = mu(x) where