Theoretical question related to a differential equation, with attached initial value problem

by dalta   Last Updated July 12, 2019 10:20 AM - source

It is my first week dealing with Differential Equations, and I am lost at the following question:

Show that the equation $P(x,y)dx+Q(x,y)dy=0$ has an integrating factor of type $μ\frac{y}{x}$ if and only if the expression $x^2\frac{P_y-Q_x}{xP+yQ}$ depends on $\frac{y}{x}$ only.

Relying on the above, resolve the following initial value problem, including its domain of definition: $(4x^5y^2+x^4e^x+2x^3e^x-\frac{3x}{y^2})dx+(4x^6y+\frac{2x^4e^x}{y})dy=0$

$y(1)=-\sqrt\frac{e}{2}$



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