Fill in the blank or answer true/false The only solution of y\" + x^2 y = 0, y(0

Question

Fill in the blank or answer true/false The only solution of y" + x^2 y = 0, y(0) = y'(0) = 0, is ____ If two differentiable functions f_1(x) and f_2(x) are linearly in dependent on an interval, then W(f_1(x), f_2(x)) notequalto 0 for at lest one point in the interval. Two functions f_1(x) and f_2(x) are linearly independent on an interval, f one is not a constant of the other ____. Two solutions y_1 and y_2 of y" + y' + y = 0 are linear dependent if W(y_1, y_2) = 0 for every real value x _____. A constant multiple of a solution of a differential equation is also a solution ____. A fundamental set of two solution of (x-2) y" + y = 0 exists on any interval not containing the point _____. For the method of undetermined coefficients, the assumed form of the particular solution y_p for y" - y = 1 + x e^x is ____ A differential operator that annihilates e^2x (x + sin x) is ____

Explanation / Answer

here questiion no.2,3,4,5 are true statement

because Let f and g be differentiable on [a,b]. If Wronskian W(f,g)(t0)  is nonzero for some t0 in [a,b] then f and g are linearly independent on [a,b].  If f and g are linearly dependent then the Wronskian is zero for all t in [a,b].

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