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When faced with a differential equation that does not initially appear to be easily solvable, the method of substitution can be a very effective tool. This technique involves replacing a part of the original equation with a new variable to simplify the equation to a more recognizable form.

The aim is to transform the given differential equation into one for which we already know the solution method. For instance, in many cases, the goal is to produce a first order linear differential equation of the form \(\frac{dy}{dt}=ky\), where \(k\) is a constant.

By introducing a new variable, say \(y\), we essentially reframe the problem, which can then be solved using standard techniques like separation of variables or integrating factors. Once the solution for \(y\) is found, we use our substitution to express the solution in terms of the original variable.

First order linear differential equations are a commonly encountered type of differential equation. These equations have the standard form \(\frac{dx}{dt}+p(t)x=q(t)\), where \(p(t)\) and \(q(t)\) are functions of \(t\), and the solution involves finding the function \(x(t)\).

One powerful method to solve these equations is by using an integrating factor, which is a function that, when multiplied by the equation, allows it to be written in a form that can be integrated directly. However, when the equation is already in the form of \(\frac{dx}{dt}=kx\), with \(k\) being a constant, solving it becomes even simpler as it then involves direct integration.

Integration is a fundamental concept in calculus, often thought of as the inverse operation to differentiation. It is the process of finding the area under a curve or, in other words, determining the antiderivative or indefinite integral of a function.

The general process of integrating a function involves finding a function \(F\) such that \(F'(x) = f(x)\), where \(f(x)\) is the given function to be integrated. The resulting function \(F\) includes a \(+ C\), representing the constant of integration, which is determined based on initial conditions or boundary values provided with the problem.

When we find the indefinite integral of a function, we introduce the constant of integration, usually denoted as \(C\). This constant represents the infinite number of antiderivatives that a function can have since the derivative of a constant is zero.

The value of \(C\) is determined by initial conditions or boundary conditions given with the problem. These conditions are specific values of the function or its derivatives at certain points. By applying these conditions, we can solve for the constant \(C\), providing us with the particular solution to the differential equation suited to the problem's context.