3.35 Solution of simple harmonic motion

Let

$$f(x)=A\cos\beta x+{{B}\over{\beta}}\sin\beta x$$

then we have

$$f^{\prime}(x)=-\beta A\sin\beta x+B\cos\beta x$$

$$f^{\prime\prime}(x)=-\beta^{2}A\cos\beta x-{{B}{\beta}}\sin\beta x$$

so $f(0)=A$, $f^{\prime}(0)=B$ and

$$f^{\prime\prime}(x)=-\beta^{2}f(x)=-{{k}\over{m}}f(x).$$

Also, $f$ is periodic since

$$f(x+2\pi/\beta)=A\cos\beta(x+2\pi/\beta)+{{B}\over{\beta}}\sin\beta(x+2\pi/\beta)$$

$$=A\cos\beta x+{{B}\over{\beta}}\sin\beta x=f(x).$$