Question

The differential equation of $y = e^x (a \cos x + b \sin x)$ is

• A. $\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}-y=0$
• B. $\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+2y=0$
• C. $\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+y=0$
• D. $\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0$

Hint:

Calculus Tip: When forming a differential equation, the order of the resulting highest derivative will always equal the number of arbitrary independent constants in the original equation (here, $a$ and $b$ mean we differentiate twice).
• Always look to substitute $y$ or $y'$ back into the differentiated equations to quickly eliminate constants.

Answer 1

The Correct Option is D

Concept: Calculus - Formation of Differential Equations (Eliminating Arbitrary Constants).

Step 1: Differentiate the given equation once. The given equation is $y = e^x(a\cos x + b\sin x)$. Differentiate with respect to $x$ using the product rule ($\frac{d}{dx}[uv] = u'v + uv'$): $\frac{dy}{dx} = e^x(a\cos x + b\sin x) + e^x(-a\sin x + b\cos x)$.

Step 2: Simplify by substituting the original function. Notice that the first term in our derivative, $e^x(a\cos x + b\sin x)$, is exactly equal to our original function $y$. Substitute $y$ back into the equation: $\frac{dy}{dx} = y + e^x(b\cos x - a\sin x)$. Let's call this Equation (i).

Step 3: Differentiate a second time. Differentiate Equation (i) with respect to $x$ to get the second derivative: $\frac{d^2y}{dx^2} = \frac{dy}{dx} + e^x(b\cos x - a\sin x) + e^x(-b\sin x - a\cos x)$.

Step 4: Eliminate constants using previous equations. Rearrange Equation (i) to isolate the exponential term: $e^x(b\cos x - a\sin x) = \frac{dy}{dx} - y$. Also, notice that the last term, $e^x(-b\sin x - a\cos x)$, is exactly the negative of our original function $y$, so it equals $-y$. Substitute both of these findings into our second derivative equation:
$\frac{d^2y}{dx^2} = \frac{dy}{dx} + \left(\frac{dy}{dx} - y\right) - y$.

Step 5: Rearrange into the final standard form. Combine the like terms on the right side of the equation: $\frac{d^2y}{dx^2} = 2\frac{dy}{dx} - 2y$. Move all terms to the left side to match the standard format of a differential equation: $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0$. $$ \therefore \text{The differential equation is } \frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=0. $$