How many common zeros in two L I Solutions of the diff. Eq. $\frac{d^2y}{dx^2} + a_1\frac{dy}{dx} +a_2y = 0$

If $y_1$ and $y_2$ are two linearly independent solutions of the differential equation

$$\frac{d^2y}{dx^2} + a_1\frac{dy}{dx} +a_2y = 0$$ where $a_1$ and $a_2$ are constants.

then y1 and y2 have

A) Odd number of common zeros

B) Exactly one common zero

C) No common zeros

D) At most two common zeros

Plz help me to solve this problem.

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If the two solutions had a common zero at a point $x_0$, than every solution would have a zero at $x_0$, because every solution can be written as a linear combination of two independent solutions. However, the initial value problem with $y(x_0)=1$, $y'(x_0)=0$, has a unique solution, so this would give a contradiction.

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Let $m^2+a_1m+a_2=0$ be characteristic equation of the differential equation so we know that :

• (When $a_1^2-4a_2 > 0$) There are two distinct real roots $r_1, r_2$. And so our linear independent solutions are $y_1=\exp(r_1x)$ and $y_2=\exp(r_2x)$.

• (When $a_1^2-4a_2 < 0$) There are two complex conjugate roots $r = λ ± μi$. Or our linear independent solutions are $y_1=\exp(λx)\sin(μx)$ and $y_2=\exp(λx)\cos(μx)$.

And

• (When $a_1^2-4a_2 < 0$) There is one repeated real root r. So the linear independent solutions are $y_1=\exp(rx)$ and $y_2=x\exp(rx)$.

Now it is clear that in what cases we are; the answer of the problem is C.

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