Find the particular solution of the differential equation \(\frac{dy}{dx}\)+2x=5 given that y=5, when x=1.

(a) y=5x+x^2+1

(b) y=x-x^2+4

(c) y=5x-x^2+1

(d) y=5x-x^2

I have been asked this question by my school principal while I was bunking the class.

Question is from Methods of Solving First Order & First Degree Differential Equations in chapter Differential Equations of Mathematics – Class 12

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Correct choice is (c) y=5x-x^2+1

Best explanation: Given that, \(\frac{dy}{dx}+2x=5\)

\(\frac{dy}{dx}=5-2x\)

Separating the variables, we get

dy=(5-2x)dx

Integrating both sides, we get

\(\int dy=\int 5-2x \,dx\)

y=5x-x^2+C –(1)

Given that, y=5, when x=1

⇒5=5(1)-(1)^2+C

∴C=1

Substituting value of C to equation (1), we get

y=5x-x^2+1 which is the particular solution of the given differential equation.