Paper 2, Section I, B

Differential Equations | Part IA, 2018

Show that for given P(x,y),Q(x,y)P(x, y), Q(x, y) there is a function F(x,y)F(x, y) such that, for any function y(x)y(x),

P(x,y)+Q(x,y)dydx=ddxF(x,y)P(x, y)+Q(x, y) \frac{d y}{d x}=\frac{d}{d x} F(x, y)

if and only if

Py=Qx\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}

Now solve the equation

(2y+3x)dydx+4x3+3y=0(2 y+3 x) \frac{d y}{d x}+4 x^{3}+3 y=0

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