Question

The differential equation having \( y = (a + b)e^{cx+d} \) as its general solution, where \( a, b, c, d \) are arbitrary constants, is

• A. \( y^{(4)} + 3y^{(3)} + 6y^{(2)}y^2 + y = 0 \)
• B. \( y^{(3)} + 4yy' + 6y^{(1)} + 12y = 0 \)
• C. \( y^{(1)} - y = 0 \)
• D. \( y^{(2)} - (y^{(1)})^2 = 0 \)

Hint:

When the general solution involves an exponential function, the corresponding differential equation is often of second order.

Answer 1

The Correct Option is D

The given differential equation corresponds to the form of an exponential function, which typically produces a second-order differential equation of the form: \[ y^{(2)} - (y^{(1)})^2 = 0 \] Therefore, the correct equation is: \[ y^{(2)} - (y^{(1)})^2 = 0 \] \[ \boxed{y^{(2)} - (y^{(1)})^2 = 0} \]