Consider the homogeneous 2nd order linear ODE \frac{d^2y}{dx^2} + 4y =0 .The general solution can be written as

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The problem asks for the general solution to the homogeneous second-order ODE y'' + 4y = 0.

First, I identify the characteristic equation associated with this linear constant-coefficient ODE: r^2 + 4 = 0.

Solving for r gives r = ±2i, which are purely imaginary roots.

When the characteristic roots are purely imaginary, the general real-valued solution is a linear combination of sine and cosine terms with the corresponding angular frequency equal to the imaginary part of the roots. Here, the frequency is 2, so the basis functions are cos(2x) and sin(2x).

Thus, the general solution can be written as y(x) = A cos(2x) + B sin(2x), where A and B are arbitrary constants. This form captures all possible solutions to the ODE, since any solution can be represented as a linear combination of the two independent solutions cos(2x) and sin(2x).

If one considers alternative representations, sometimes you might see equivalents like y(x) = C1 cos(2x) + C2 sin(2x), which is the same family with different constant names. No exponential terms remain because the roots are imaginary, not real.

In the given answer option, y(x) = A cos(2x) + B sin(2x) matches this standard general solution form exactly, confirming its correctness in the usual real-valued representation. Since there are no other answer choices provided to compare, there isn’t an alternate option to evaluate against in this context.

Note: If initial conditions were supplied, A and B would be determined from y(0) and y'(0), but those conditions are not given here.

PHAS0009_24-25 5.3 Quiz 5 [0.25 hrs]

Consider the homogeneous 2nd order linear ODE \frac{d^2y}{dx^2} + 4y =0 .The general solution can be written as

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