Q. 175.0( 1 Vote )

# Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2(x + 1) = 2e2x.

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 + cex ……(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 + 2ex

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