Let's Solve An Interesting Differential Equation

This paragraph introduces the topic of solving a non-homogeneous differential equation, specifically \( y' + y = x^2 e^x \). The speaker explains the process of finding the general solution by first solving the homogeneous case and then adding a particular solution. The method of considering a solution in the form \( y = e^{rx} \) is introduced, leading to the characteristic equation \( r + 1 = 0 \). The homogeneous solution is identified as \( y_h = C_1 e^x \), where \( C_1 \) is a constant.

The speaker delves into the variation of parameters method for finding a particular solution to the given differential equation. They propose a solution form \( y_p = v(x)e^x \), where \( v(x) \) is a function to be determined. By applying the product rule and simplifying, the equation \( v'e^x = x^2 e^x \) is derived. The solution involves isolating \( v' \) and integrating to find \( v(x) \), which leads to a particular solution involving terms like \( 12x^2 e^{2x} \), \( -12xe^{2x} \), and \( \frac{1}{4}e^{2x} \), plus an integration constant.

The final paragraph wraps up the video by presenting the general solution to the differential equation, which is the sum of the homogeneous solution \( C_1e^x \) and the particular solution involving the terms found using the variation of parameters method. The speaker reminds viewers to engage with the content by commenting, liking, and subscribing, and concludes with well-wishes for the audience's safety and health until the next video.