Find the differential equation satisfied by y=sin(x+c) and…

Find the differential equation satisfied by $y=\sin(x+c)$ and solve that differential equation.

Step 1: Eliminate $c$

Differentiating $y=\sin(x+c)$ twice gives $y'=\cos(x+c)$ and $y''=-\sin(x+c)=-y$, so

\[\boxed{\,y''+y=0.}\]

Step 2: Solve the characteristic equation

Assuming $y=e^{rx}$ leads to $r^2+1=0$, so $r=\pm i$, giving the general real solution

\[y(x)=C_1\cos x+C_2\sin x.\]

Step 3: Amplitude–phase form

With $R=\sqrt{C_1^2+C_2^2}$ and a suitable phase $\varphi$, this can be written as $y=R\sin(x+\varphi)$. The special case $R=1,\ \varphi=c$ gives exactly $y=\sin(x+c)$.