Linear Second Order ODE/y'' + y = 0/y(0) = 2, y'(0) = 3

Theorem

The second order ODE:

$(1): \quad y + y = 0$

with initial conditions:

$\map y 0 = 2$

$\map {y'} 0 = 3$

has the particular solution:

$y = 3 \sin x + 2 \cos x$

Proof

From Linear Second Order ODE: $y + y = 0$, the general solution of $(1)$ is:

$y = C_1 \sin x + C_2 \cos x$

Differentiating with respect to $x$:

$y' = C_1 \cos x - C_2 \sin x$

Thus for the initial conditions:

\(\ds \map y 0\)

\(=\)

\(\ds C_1 \sin 0 + C_2 \cos 0\)

\(\ds \)

\(=\)

\(\ds 2\)

\(\ds \leadsto \ \ \)

\(\ds C_1 \times 0 + C_2 \times 1\)

\(=\)

\(\ds 2\)

Sine of Zero is Zero, Cosine of Zero is One

\(\ds \leadsto \ \ \)

\(\ds C_2\)

\(=\)

\(\ds 2\)

and:

\(\ds \map {y'} 0\)

\(=\)

\(\ds C_1 \cos 0 - C_2 \sin 0\)

\(\ds \)

\(=\)

\(\ds 2\)

\(\ds \leadsto \ \ \)

\(\ds C_1 \times 1 - C_2 \times 0\)

\(=\)

\(\ds 3\)

Sine of Zero is Zero, Cosine of Zero is One

\(\ds \leadsto \ \ \)

\(\ds C_1\)

\(=\)

\(\ds 3\)

Hence the result.

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.15$: The General Solution of the Homogeneous Equation: Example $1$