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(d^(2)y)/(dx^(2))+y=0

$3$ . $\frac{d^{2} y}{d x^{2}}+y=0$

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Evaluate integers raised to rational exponents

Full solution

Q.

3

.

\frac{d^{2} y}{d x^{2}}+y=0

Identify Type of Equation: Identify the type of differential equation. This is a second-order linear homogeneous differential equation with constant coefficients.
Guess Solution Form: Guess a solution of the form $y = e^{rx}$ . $\newline$ We'll substitute this into the differential equation to find the value of $r$ .
Substitute into Equation: Substitute $y = e^{rx}$ into the equation. $\frac{d^2y}{dx^2} + y = r^2 \cdot e^{rx} + e^{rx} = 0$
Factor Out: Factor out $e^{rx}$ . $\newline$ $e^{rx} * (r^2 + 1) = 0$
Set Equal to Zero: Set the expression in parentheses equal to zero. $r^2 + 1 = 0$
Solve for r: Solve for r. $\newline$ $r^2 = -1$ $\newline$ $r = \pm i$
Write General Solution: Write the general solution. $\newline$ $y = C_1 \cdot e^{i x} + C_2 \cdot e^{-i x}$ $\newline$ But $e^{i x}$ and $e^{-i x}$ can be expressed using Euler's formula as $\cos(x)$ and $\sin(x)$ .
Express Using Trig Functions: Express the solution using trigonometric functions. $\newline$ $y = C_1 \cdot \cos(x) + C_2 \cdot \sin(x)$

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