Let be two distinct roots of where a, b, c are three real constants and . Then, is also a root of the same equation, if
a + b = c
b + c = a
c + a = b
c = a
If are solutions of the differential equation
where a0, a1 and a2 are real constants, then which of the following is/are always true?
is a solution, where A and B are real constants
is a solution, where A is a real constant
is a solution, where A is a real constant
is a souton, where A and B are real constants
A.
is a solution, where A and B are real constants
B.
is a solution, where A is a real constant
D.
is a souton, where A and B are real constants
(a) Let
Consider the Wronskian of f(x) and g(x),
Thus, the functions are linearly independent. So, the general solution of given differential equation is given by , where A and B are real constants.
[ if y1 and y2 are linearly independent solutions of the differential equation ay'' + by' + c = 0, then the general solution is y = c1y1 + c2y2, where c1 and c2 are constants]
Hence, option (a) is true.
(b) Let
which is in the form of general solution.
Hence, option (b) is true
(c) Let y = , which cannot be expressed in the form of general solution.
(d) Let y
which is in the form of general solution.
Hence, option (d) is true.
In a , a, b, c are the sides of the triangle opposite to the angles A, B, C, respectively. Then, the value of a3sin(B - C) + b3sin(C - A) + c3sin(A - B) is equal to
0
1
3
2