Verify that y=sinx, y=cosx, y=e^i x, and y=e^-i x are all solutions of y^''=-y.

Name: Problem 3, Chapter 8: Mathematical Methods in the Physical Sciences
Uploaded: 2022-01-05T12:42:08-08:00
Duration: 4 min 24 s
Channel: Aditya Jain
Description: Verify that y=sinx, y=cosx, y=e^i x, and y=e^-i x are all solutions of y^''=-y.

step 1 All right, so we have to solve this question. It says, verify that y equals sine x, y equals cos

Verify that y=sinx, y=cosx, y=e^i x, and y=e^-i x are all...

Key Concepts

Verification by Substitution

This concept involves taking a candidate function and its derivatives and substituting them back into the differential equation to ensure that the equation is satisfied identically. In the context of differential equations, this method is a straightforward way to confirm that a function is indeed a solution by showing that both sides of the equation match upon substitution.

Linear Homogeneous Differential Equations

A linear homogeneous differential equation has the form where every term is a linear function of the unknown function and its derivatives, and there is no free (non-zero) term. The equation y'' = -y is a second-order linear homogeneous differential equation with constant coefficients. Understanding these types of equations is crucial, as they often have solutions that can be expressed in terms of exponential and trigonometric functions, and they obey the principle of superposition.

Euler's Formula

Euler's formula, which states that e^(ix) = cos x + i sin x, provides a deep connection between complex exponentials and trigonometric functions. This concept is essential while verifying solutions like y = e^(ix) and y = e^(-ix) of the differential equation, as it shows that these complex exponentials naturally encapsulate the sine and cosine functions, which are also solutions of the equation.

Differentiation Rules for Trigonometric and Exponential Functions

Differentiation rules for trigonometric and exponential functions are fundamental in obtaining the derivatives needed for the verification process. Knowing that the derivative of sin x is cos x, the derivative of cos x is -sin x, and the derivatives of e^(ix) and e^(-ix) involve multiplying by the constant from the exponent (including the imaginary unit i for complex exponentials) allows a systematic approach to showing that the second derivatives indeed yield -y.

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