Suppose that y1 and y2 are solutions of the second order DE y\'\' + y\'p(t) + y

Question

Suppose that y1 and y2 are solutions of the second order DE y'' + y'p(t) + y q(t) = 0.
Show that C1y1 + C2y2 is also a solution where C1 and C2 are arbitrary constants. This
fact is called the principle of superposition and the expression C1y1 + C2y2 is called a linear
combination of y1 and y2.

6b) The DE in the problem 6a was of a special type. Can you think of DE where the
principle of superposition does not hold? (In other words, can you think of DE where a
linear combination of two solutions is not a solution?).

Explanation / Answer

y'' + y'p(t) + y q(t) = 0 if y1,y2 are the solutions of the above DE ==> y1"+y1'p(t)+y1q(t) =0 ---------->1 ==> y2"+y2'p(t)+y2q(t) =0 ---------->2 eq 1 x C1 ==> C1y1"+C1y1'p(t)+C1y1q(t) =0 ---------->3 eq 2 x C2 ==> C2y2"+C2y2'p(t)+C2y2q(t) =0 ---------->4 Add the eqs 3 and 4 ==> C1y1"+C2y2"+C1y1'p(t)+C2y2'p(t)+C1y1q(t)+C2y2q(t) =0 ==> (C1y1+C2y2)"+(C1y1+C2y2)'p(t)+C1y1+C2y2)q(t) =0 -------------->5 from equation 5 we can conclude that C1y1+C2y2 is also a solution of the given DE

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