91Ó°ÊÓ

First-order linear differential equations are a fundamental type of differential equation.
These equations involve derivatives of the first order (meaning we only deal with the first derivative). The standard form of such an equation is \(a(x) y' + b(x) y = c(x)\), where:\

• \(a(x)\), the coefficient of \(y'\), is a function of \(x\).
• \(b(x)\), the coefficient of \(y\), is another function of \(x\).
• \(c(x)\) is the non-homogeneous part, also depending on \(x\).

These equations are important because they appear frequently in various natural and applied sciences, such as engineering, physics, and biology. Solving them generally involves finding a function \(y(x)\) that satisfies this equation.
Understanding the structure of the differential equation is crucial as it informs us about the possible methods we can use for solving, such as utilizing an integrating factor. First-order linear differential equations can be solved systematically and provide a stepping stone to more complex mathematical concepts.

The integrating factor in solving first-order linear differential equations is like a magic key.
It's a mathematical function we multiply through the entire differential equation to turn it into a form that’s easier to handle.When dealing with an equation of the form \(a(x) y' + b(x) y = c(x)\), you can find the integrating factor \(\mu(x)\) by using the formula:\[\mu(x) = e^{\int \frac{b(x)}{a(x)} \, dx} \]This interesting tool allows us to convert the left side of the equation into a single derivative. So, when the integrating factor is multiplied across, the left side becomes of the form \(\frac{d}{dx}(\mu(x)y)\), making the equation set up for easy integration.
With our given equation, the integrating factor was calculated as \(1+x\), a simple yet powerful tweak that transformed the problem into a more manageable form.

Verifying the solution of a differential equation ensures that the final function we find is indeed correct.
This step is vital to confirm that every part of our solution satisfies the original differential equation given in the problem.After solving the equation for \(y(x)\), you verify by differentiating your solution and substituting back into the original equation:\[ (1+x) y' + y = \sqrt{x} \]Substitute \(y(x)\) back in from your solution and check that both sides are indeed equal.
If they are, it is a strong confirmation that the solution is correct. This step highlights the precision needed in solving differential equations and showcases how beautifully mathematics allows us to check our work systematically.

Mathematical integration is essential in solving differential equations, especially in the final stages where it helps find the solution function from its derivative.
In our context, it was used after rewriting the differential equation using the integrating factor. Here’s the general process:

• We begin by setting up the integral on both sides of the equation.
• The left side of the equation integrates to give back the function expression \((1+x)y\).
• On the right side, we expand and calculate the integral of expressions like \(\sqrt{x}(1+x)\), which breaks into two separate terms.

In our exercise, integration gave terms like \(\frac{2}{3}x^{3/2}\) and \(\frac{2}{5}x^{5/2}\), showcasing how integrations can unravel complex expressions.
Including the constant of integration is a crucial part of this process, representing an infinite set of solutions aligned with the differential equation. This element adds depth and completeness to the finalized function \(y(x)\).