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1) First why is the general form of the solution to linear homogenous differential equations, with non-equal and real roots to the complementary equation, of the forum

[tex]y = {e^{{\alpha _1}x}} + {e^{{\alpha _2}x}} + \cdot \cdot \cdot + {e^{{\alpha _n}x}}[/tex]

And real and equal roots to the complementary equation

[tex]y = {e^{{\alpha _1}x}} + x{e^{{\alpha _2}x}} + \cdot \cdot \cdot + {x^{n - 1}}{e^{{\alpha _n}x}}[/tex]

It seems that the resources on this subjects simply tell you this is the form of the solution with very little explanation of why this is known to be the form of the solution. If anyone can give me some insight into this I would appreciate it. Thanks

2) The second question is how do we know that the general solution to the nth order non-homogenous differential equation is: [tex]y(t) = {y_p}(t) + {y_h}(t)[/tex]