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In the realm of differential equations, particularly first-order linear differential equations, the "general solution" plays a pivotal role. It represents the broadest form of solutions applicable to a differential equation. For a first-order linear differential equation such as

• \( y' = a(x) y + b(x) \)

to express its general solution, you want to focus on finding an expression that satisfies the equation for arbitrary conditions. In the given exercise, we were tasked with finding the general solution for the non-homogeneous differential equation \( y' = 3x + e^x \). Here, the approach involves integrating each part separately. To derive the general solution:

• First, you separate the equation into its individual components \(3x\) and \(e^x\).
• Then, you integrate each component with respect to \(x\).

After performing these steps, you recombine the integrated expressions and add the constant \(C\), which accounts for any initial conditions that may apply later. Thus, in this context, the general solution becomes \( y = \frac{3}{2}x^2 + e^x + C \). This formulation is critical as it encapsulates all possible solutions across different scenarios.

Integration is a fundamental concept in calculus, serving as the reverse process of differentiation. It is essential when solving first-order linear differential equations. Within this context, integration transforms a differential equation into its general solution. Specifically, in our given problem, we encounter two distinct integrands: a polynomial term \(3x\) and an exponential term \(e^x\).

• For the polynomial \(3x\), integration proceeds by applying the power rule for integration: \(\int 3x \, dx = \frac{3}{2} x^2\). This method involves raising the power of \(x\) by one and then dividing by this new exponent.
• For the exponential function \(e^x\), integration is straightforward since the integral of \(e^x\) is itself: \(\int e^x \, dx = e^x\).

By understanding how to integrate these components, you can effectively piece together the solution of our differential equation. Ultimately, integration used in this manner provides the building blocks to construct the general solution \(y = \frac{3}{2}x^2 + e^x + C\), thereby solving the differential equation.

The term "non-homogeneous differential equation" refers to a differential equation that includes an external function or forcing term. It makes these equations distinct from homogeneous ones. For our specific problem, we have:

• The given equation is \(y' = 3x + e^x\); the right-hand side features functions of \(x\), namely \(3x\) and \(e^x\).

These terms are called non-homogeneous because they introduce additional variation that is not simply a scalar multiple of the unknown function or its derivatives.To solve such equations, you typically separate them into complementary and particular solutions.

• The complementary solution solves the associated homogeneous equation (obtained by setting the non-homogeneous part to zero).
• The particular solution addresses the specific non-homogeneity introduced by the external function.

For this instance, the differential equation's structure allows direct integration since it is relatively simple. The combination of these strategies leads us to the general solution \( y = \frac{3}{2}x^2 + e^x + C \). By apprehending these procedures, solving non-homogeneous differential equations becomes an insightful process to address real-world scenarios with external influences.