What is the solution for x(dy)/(dx)=x^2+2x-3 ?

The solution for x(dy)/(dx)=x^2+2x-3 is y=(x^2)/2+2x-3ln(x)+c_{1}

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x(dy)/(dx)=x^2+2x-3

x \frac{d y}{d x} = x^{2} + 2 x - 3

y = \frac{x ^{2}}{2} + 2 x - 3 ln (x) + c_{1}

Solution steps

x \frac{d y}{d x} = x^{2} + 2 x - 3

Substitute

\frac{d y}{d x}

with

y^{'} (x)

x y^{^{'}} (x) = x^{2} + 2 x - 3

Isolate

y^{'} (x) : y^{'} (x) = \frac{x ^{2} + 2 x - 3}{x}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

y = \int \frac{x ^{2} + 2 x - 3}{x} d x

\int \frac{x ^{2} + 2 x - 3}{x} d x = \frac{x ^{2}}{2} + 2 x - 3 ln (x) + c_{1}

y = \frac{x ^{2}}{2} + 2 x - 3 ln (x) + c_{1}

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What is the solution for x(dy)/(dx)=x^2+2x-3 ?
The solution for x(dy)/(dx)=x^2+2x-3 is y=(x^2)/2+2x-3ln(x)+c_{1}