Solve First Order Linear Differential Equation

Solve the differential equation: $\frac{dy}{dx} + 5y = e^{2x}$.

First-order linear differential equations are equations of the form dy/dx + P(x)y = Q(x), where P and Q are functions of x. The given problem is already in this standard form with P(x) equal to 5 and Q(x) equal to e to the power of $2x$. To solve these equations, an integrating factor is used, which is derived from the function P(x). The integrating factor simplifies the equation into a form that can be easily integrated. This technique highlights the importance of transforming differential equations to more workable forms using algebraic manipulations and multipliers, akin to other techniques used across various mathematical problems, like completing the square or rationalizing the denominator. The knowledge of linear differential equations not only applies to solving theoretical problems, but also finds applications in various fields like physics, engineering, and applied mathematics where such models help understand natural phenomena and solve engineering problems involving rates of change. Using this systematic approach aids in understanding the behavior of solutions based on initial conditions and provides insights into the stability of systems described by differential equations. In learning this topic, gaining proficiency in recognizing the standard form, finding and applying the integrating factor, and performing the integration to solve for the function y(x), are the key skills developed.

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