# 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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## Non bounded ODE solution

Updated January 26, 2019 01:20 AM