The differential equation representing the curve \(y = e^{2x}(a + bx)\), where \(a, b\) are arbitrary constants is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0\)
There are two arbitrary constants, so the equation is of order 2; it is obtained by differentiating twice and eliminating a and b.
\(y = e^{2x}(a + bx)\)
\(y' = 2e^{2x}(a + bx) + be^{2x}\)
\(y' = 2y + be^{2x}\)
So \(be^{2x} = y' - 2y\). Differentiating \(y' = 2y + be^{2x}\) again:
\(y'' = 2y' + 2be^{2x}\)
\(y'' = 2y' + 2(y' - 2y)\)
\(y'' = 4y' - 4y\)
\( \displaystyle \frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0 \)