Moderate# First Order Linear Equation and Superposition Principles

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Consider the first order linear equation:

$$y'+2y=x^2$$

According to the **superposition principles**, we can first find a particular solution $y_p$ of the given equation and then solve the homogeneous equation:

$$y'+2y=0$$

The homogeneous equation can be solved , for example, using the characteristic equation $r+2=0$ to find $y_h=Ce^{-2x}$. (Of course, you can also use the technique for separable equations to find $y_h$). The original equation can then be solved.

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