If y =e3x +x2 , then prove that d2 y÷ dx2 -2 =9(y-x2)

1.	If y =e3x +x2 , then prove that d2 y÷ dx2 -2 =9(y-x2)
Answer» y = e^3x + x² ⇒ \(\frac{dy}{dx}\) = 3e^3x + 2x ⇒ \(\frac{d^2y}{dx^2}\) = 9e^3x + 2 ⇒ \(\frac{d^2y}{dx^2}\) - 2 = 9e^3x ⇒ \(\frac{d^2y}{dx^2}\) - 2 = 9(y - x²) (∵ y = e^3x + x² ⇒ e^3x = y - x²) Hence, proved.

Answer»

y = e^3x + x²

⇒ \(\frac{dy}{dx}\) = 3e^3x + 2x

⇒ \(\frac{d^2y}{dx^2}\) = 9e^3x + 2

⇒ \(\frac{d^2y}{dx^2}\) - 2 = 9e^3x

⇒ \(\frac{d^2y}{dx^2}\) - 2 = 9(y - x²) (∵ y = e^3x + x² ⇒ e^3x = y - x²)

Hence, proved.

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If y =e3x +x2 , then prove that d2 y÷ dx2 -2 =9(y-x2)

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