The expression y = α · sin x + β · cos x is a solution of the ordinary differential equations if and only if (α, β) = (0, 0).
When a given equation is a solution of an ordinary differential equation
According to the statement, we must find in what conditions a given expression may be a solution of an ordinary differential equation. Then, first and second derivatives of the equation are:
y' = α · cos x - β · sin x (1)
y'' = - α · sin x - β · cos x (2)
Then, we substitute on the ordinary differential equation:
(- α · sin x - β · cos x) + (α · cos x - β · sin x) = 0
And by algebraic handling we simplify the resulting expression:
- (α + β) · sin x + (α - β) · cos x = 0
Where each coefficient represents a constant of a linear combination:
α + β = 0
α - β = 0
Then, the solution of the system of linear equations is (α, β) = (0, 0). The expression y = α · sin x + β · cos x is a solution of the ordinary differential equations if and only if (α, β) = (0, 0).
Remark
The statement is incomplete and complete form cannot be found, then we decided to create a new statement:
Please prove that y = α · sin x + β · cos x is the solution of the differential equation d²y / dx² + dy /dx = 0 where the following condition is observed, if and only if α = β = 0.
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