[Math] The differential equation $(D^2+1)y=0$

calculusintegrationordinary differential equations

The differential equation $(D^2+1)y=0$, $D$ is differential operator and $y(0)=1 , y(\pi)=0$
has

A. unique solution

B. Single infinite family of solutions

C. No solution

D. Doubly infinite family of solutions

I obtained solution as $y(x)=c_1 \cos x+c_2 \sin x$ .

How do I choose correct option?

Best Answer

The equation need not be solved given what the question actually wants. It does not require the equation to be solved.

The given differential equation is of $2$nd order and hence its general solution, as you have written, contains two arbitrary constants $c_1$ and $c_2$.

To find the $2$ constants definitively, you need $2$ boundary conditions. And those two conditions are given as follows:

$$y(0)=1$$ $$y(\pi)=0$$

So you can determine the $2$ constants and what you then have is a unique solution.

Hence the correct answer should have been in general OPTION A.

BUT observe that here the $2$ conditions do not allow the $2$ constants to be determined. Moreover it causes an indeterminacy in the value of $c_1$. Hence this equation in particular has no solution.

Hence the correct answer is OPTION C.

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