From x(0)=x0=c1 we see that x(t)=x0 cosωt+c2 sinωt and x^'(t)=-x0 sinωt+c2 ωcosωt . Then x^'(0)=

step 1 Solving part of this problem, so here I can write the value of L is equal to A sine omega T v is

From x(0)=x0=c1 we see that x(t)=x0 cosωt+c2 sinωt and...

From $x(0)=x_{0}=c_{1}$ we see that $x(t)=x_{0} \cos \omega t+c_{2} \sin \omega t$ and $x^{\prime}(t)=-x_{0} \sin \omega t+c_{2} \omega \cos \omega t .$ Then $x^{\prime}(0)=x_{1}=c_{2} \omega$ implies $c_{2}=x_{1} / \omega .$ Thus $$x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t.$$

Key Concepts

Application of Trigonometric Functions in Differential Equations

Trigonometric functions, like sine and cosine, naturally arise as solutions to differential equations describing oscillatory phenomena. Their properties make them ideal for representing periodic behavior, with coefficients determined by initial conditions reflecting the system's state at a specific time.

Second Order Homogeneous Linear Differential Equations

These are differential equations involving a second derivative and no terms that depend on the independent variable explicitly, allowing solutions to be expressed as a linear combination of two independent functions. Typically, the general solution is derived using the characteristic equation, leading to solutions in the form of exponential, sine, and cosine functions depending on the nature of the roots.

Simple Harmonic Motion

Simple harmonic motion is a type of periodic oscillation characterized by a restoring force proportional to displacement, leading to differential equations whose solutions are sine and cosine functions. This concept is central in understanding systems that exhibit repetitive, oscillatory behavior, such as springs and pendulums.

Initial Value Problems

An initial value problem involves finding a specific solution to a differential equation by using initial conditions such as the value of the function and its derivatives at a particular point. This process determines the constants in the general solution, ensuring that the solution satisfies the specific starting conditions of the physical or mathematical system.

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