91Ó°ÊÓ

The general solution of a differential equation provides all possible solutions for the equation. It encapsulates the complete set of answers obtained from integrating the differential equation.
For the given form of the differential equation, \( y^{\backprime}+M(x)y = N(x) \), the general solution can be expressed as:
\[ y = \frac{\backint P(x)N(x) dx + C}{P(x)} \]
where\( P(x) \) is the integrating factor.
This formula is incredibly useful because it provides a straightforward process to solve differential equations once the integrating factor is determined. It brings structure to solving what might initially seem like complex problems. Let's move to understand what an integrating factor is and how to find it.

To solve the differential equation, we first need to find the integrating factor, denoted as \( P(x) \). The integrating factor is a special function that simplifies a differential equation, making it easier to solve.
In the given problem, the function \( M(x) \) is identified, and here it is \( M(x) = x^2 \). To find the integrating factor \( P(x) \), we use the formula:
\[ P(x) = e^{\backint M(x) dx} \]
For our specific \( M(x) \), we compute:
\[ P(x) = e^{\backint x^2 dx} \]
The integral of \( x^2 \) is \( \frac{x^3}{3} \), so,
\[ P(x) = e^{\frac{x^3}{3}} \]
This integrating factor plays a crucial role in simplifying the next step of solving the differential equation.

The substitution method involves substituting one expression with another to simplify integration. It's a useful technique, especially when the integral is not straightforward.
In our problem, we identify \( N(x) \) as \( x^2 \). To compute the integral \( \backint P(x)N(x) dx \) with \( P(x) \) we found earlier,
\[ \backint e^{\frac{x^3}{3}} x^2 dx \]
We use substitution to tackle this integral. Let \( u = \frac{x^3}{3} \), so, \( du = x^2 dx \). Now, the integral becomes:
\[ \backint e^u du = e^u + C = e^{\frac{x^3}{3}} + C \]
The substitution thus simplifies the integration process, helping us to find the integral more easily. This value will be crucial for the next step.

Breaking down the solution into steps helps in understanding each part of solving a differential equation:
1. **Determine the Integrating Factor, \( P(x) \)**
Identify \( M(x) \), then calculate \( P(x) \) using \( P(x) = e^{\backint M(x) dx} \). Here,
\[ P(x) = e^{\frac{x^3}{3}} \]
2. **Determine the Integral of \( P(x)N(x) \)**
Identify \( N(x) \) and integrate \( P(x)N(x) \):
\[ \backint e^{\frac{x^3}{3}} x^2 dx = e^{\frac{x^3}{3}} + C \]
3. **Divide by \( P(x) \)**
Take the result and divide by \( P(x) \):
\[ y = \frac{e^{\frac{x^3}{3}} + C}{e^{\frac{x^3}{3}}} = 1 + Ce^{-\frac{x^3}{3}} \]
Hence, the general solution is:
\[ y = 1 + Ce^{-\frac{x^3}{3}} \]
Breaking the process down makes it manageable and highlights the importance of each technique used.