Paper 2, Section I, B

Differential Equations | Part IA, 2014

Consider the ordinary differential equation

P(x,y)+Q(x,y)dydx=0.P(x, y)+Q(x, y) \frac{d y}{d x}=0 .

State an equation to be satisfied by PP and QQ that ensures that equation ()(*) is exact. In this case, express the general solution of equation ()(*) in terms of a function F(x,y)F(x, y) which should be defined in terms of PP and QQ.

Consider the equation

dydx=4x+3y3x+3y2\frac{d y}{d x}=-\frac{4 x+3 y}{3 x+3 y^{2}}

satisfying the boundary condition y(1)=2y(1)=2. Find an explicit relation between yy and xx.

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