Given slope at any point = y + 2x

⇒

⇒

We can see that it is a linear differential equation.

Comparing it with

P = – 1, Q = 2x

I.F = e^∫Pdx

= e ^{– dx}

= e ^{– x}

Solution of the given equation is given by

y × I.F = ∫Q × I.F dx + c

⇒ y × e ^{– x} = ∫ 2x × e ^{– x} dx + c

⇒ ye ^{– x} = 2∫ x × e ^{– x} dx + c

⇒ ye ^{– x} = – 2x e ^{– x –} 2 e ^{– x} + c

⇒y = – 2x – 2 + ce^x ……(1)

As the equation passing through origin,

0 = 0 – 2 + c× 1

⇒ c = 2

Putting the value of c in equation (1)

∴ y = – 2x – 2 + 2e^x