Solution of a differential equation that are $C^2(\mathbb{R} \setminus \{0 \})$, continuous on $\mathbb{R}$ and even

Question

I have to find all the solution of the differential equation $$-y''+y=0$$ that belongs in $C^2(\mathbb{R} \setminus \{0 \})$, that are even, and continuous on $\mathbb{R}$. This gives me $$ y(x)=A_-e^{-x} + B_- e^x \ \ \text{for} \ x<0$$ and $$y(x)=A_+e^{-x} + B_+ e^x \ \ \text{for} \ x>0.$$ Now i use the fact that $y$ must be continuous in $0$, this gives me $$A_- + B_- = A_+ + B_+.$$ Now i am not sure how to use the fact that $y$ must be even, this would gives me $$A_+e^{-x} + B_+ e^{x}=A_-e^{x} + B_- e^{-x}.$$ I know that i have to find $A_-= B_+$ and $B_-=A_+,$ but i'm not sure how to conclude.

Answer 1

Multiplying $A_- +B_-= A_+ + B_+$ by $e^x$ gives $A_-e^{x} +B_-e^{x}= A_+e^{x} + B_+e^{x}$ and subtracting with the second

$$A_-e^{x} +B_-e^{x}-A_+e^{-x}-B_+e^{x}=A_+e^{x} + B_+e^{x}-A_-e^{x}-B_-e^{-x}$$ $$(A_-+B_--B_+)e^{x}-A_+e^{-x}=(A_++B_+-A_-)e^{x}-B_-e^{-x}$$

Since this must hold for $x\in\mathbb{R}$, compare coefficients of $e^{x}$ and $e^{-x}$.