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Consider the first order linear equation:


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


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.

Select ALL the correct statements regarding this method.


One particular solution is obtained by the constant solution, or $y'=0$. Hence, $2y_p=x^2$ or $y_p=\cfrac{1}{2}x^2$.


One particular solution can be obtained by trying quadratic function $y_p=ax^2+bx+c$, yielding $y_p=\frac{1}{2}x^2-\cfrac{1}{2}x+\frac{1}{4}$.


The full list of solutions to $y'+2y=x^2$ is $y=Ce^{-2x}$.


The full list of solutions to $y'+2y=x^2$ is $y=Ce^{-2x}+\cfrac{1}{2}x^2$.


The full list of solutions to $y'+2y=x^2$ is $y=Ce^{-2x}+\cfrac{1}{2}x^2-\cfrac{1}{2}x+\cfrac{1}{4}$.

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