Solved Problems In Differential Equations

Solved Problems In Differential Equations-6
Furthermore, the left-hand side of the equation is the derivative of \(y\).

Furthermore, the left-hand side of the equation is the derivative of \(y\).

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It can be shown that any solution of this differential equation must be of the form \(y=x^2 C\).

This is an example of a general solution to a differential equation.

One such function is \(y=x^3\), so this function is considered a is an equation involving an unknown function \(y=f(x)\) and one or more of its derivatives.

A solution to a differential equation is a function \(y=f(x)\) that satisfies the differential equation when \(f\) and its derivatives are substituted into the equation. Note that a solution to a differential equation is not necessarily unique, primarily because the derivative of a constant is zero.

Consider the equation \(y′=3x^2,\) which is an example of a differential equation because it includes a derivative.

There is a relationship between the variables \(x\) and \(y:y\) is an unknown function of \(x\).

In fact, any function of the form \(y=x^2 C\), where \(C\) represents any constant, is a solution as well.

The reason is that the derivative of \(x^2 C\) is \(2x\), regardless of the value of \(C\).

Calculus is the mathematics of change, and rates of change are expressed by derivatives.

Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function \(y=f(x)\) and its derivative, known as a differential equation.


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