Verify that y=cosx and y=sinx are solutions of the DE y^''+y=0 . Are any of the following functi

step 1 Hello, we have to verify that y equals to cos x and y equals to sin x are the solution of differ

Verify that y=cosx and y=sinx are solutions of the DE...

Verify that $y=\cos x$ and $y=\sin x$ are solutions of the DE $y^{\prime \prime}+y=0 .$ Are any of the following functions solutions? (a) $\sin x-\cos x,(b) \sin (x+3),(\mathrm{c}) \sin 2 x .$ Justify your answers.

Key Concepts

Verification by Substitution

Verifying solutions of differential equations involves differentiating the candidate function and directly substituting it and its derivatives into the equation to see if the equality holds for all x. This straightforward method is used to confirm whether the function meets the requirements imposed by the differential equation.

Trigonometric Functions as Solutions

Trigonometric functions like sine and cosine naturally satisfy certain second order linear differential equations due to their cyclical differentiation properties. For example, differentiating sine or cosine twice typically returns the original function multiplied by a constant factor, a characteristic that makes them ideal candidates for solutions of equations like y'' + y = 0.

Differentiation Using the Chain Rule

When candidate solutions involve composed functions, such as sine or cosine of a linear function (e.g., sin(2x) or sin(x+3)), the chain rule is essential for correctly differentiating these expressions. The chain rule takes into account the derivative of the inner function, which is crucial in verifying the solution.

Second Order Linear Homogeneous Differential Equations

This concept deals with differential equations of the form y'' + p(x)y' + q(x)y = 0, where the function and its derivatives appear linearly. For constant coefficients, such equations often have solutions that can be expressed in terms of exponential, sine, and cosine functions. The structure allows one to verify candidate solutions by substituting them back into the equation.

Superposition Principle

In the context of linear homogeneous differential equations, the superposition principle states that any linear combination (with constant coefficients) of solutions is also a solution. This principle is fundamental because it enables the construction of the general solution once a set of linearly independent solutions is identified.

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