Form the differential equation of y = ae3x cos(x + b) Where y' = \(\rm \frac {dy}{dx}\) and y" = \(\rm \frac {d^2y}{dx^2}\)

Form the differential equation of y = ae3x cos(x + b) Where y' = \(\rm \frac {dy}{dx}\) and y" = \(\rm \frac {d^2y}{dx^2}\) Correct Answer y'' - 6y' + 10y = 0

Concept:

To form the differential equation of the given equation

• Differentiate the equation, the number of times as many as the constants are there.
• Find out the constants in terms of the variables.
• Substitute the variables in the original equation.

Calculation:

Given equation is y = ae3xcos(x + b)

There are two constants a and b so differentiate two times

Differentiating w.r.t x, we get

⇒ y' = 3ae^3xcos(x + b) - ae^3xsin (x + b)

⇒ y' = 3y - ae3xsin (x + b)

⇒ ae3xsin (x + b) = 3y - y'

Differentiating again w.r.t x, we get

⇒ 3ae3xsin (x + b) + ae3xcos (x + b) = 3y' - y''

⇒ 3(3y - y') + y - 3y' + y'' = 0

⇒ 9y - 3y' + y - 3y' + y'' = 0

⇒ y'' - 6y' + 10y = 0