Let (*) be a homogeneous equation of the form (*) y" + a1y' + aoy = 0 for constants a1, ao € R. To such an equation, we can also associate a polynomial x(t) = t² + a;t + ao- Suppose x(t) splits into linear factors, say x(t) = (t – a)(t - B) for a, 3 € R. Show that y = y(r) is a solution to (*) if and only if y is a solution to y" – (a + 3)y/ + a3y = 0. Next, show that (i) if a # 3, then y1 = eaz and y2 = e3z form a basis of solutions to (*); and

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.