# Use the family in Problem 1 to ﬁnd a solution of y+y''=0 that satisﬁes the boundary conditions y(0)=0,y(1)=1.

Use the family in Problem 1 to ﬁnd a solution of $y+y=0$ that satisﬁes the boundary conditions $y\left(0\right)=0,y\left(1\right)=1.$

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Ayesha Gomez

We have to find the solution of the given initial value problem,
The differential equation can be written as, $\left({D}^{2}+1\right)y=0\dots .\left(1\right)$
where $D==\frac{d}{dx},{D}^{2}==\frac{{d}^{2}}{{dx}^{2}}$
So the auxiliari equation of (1) is, ${D}^{2}+1=0\to {D}^{2}=-1\to D=+-i$
The required general solution is, $y=c1\mathrm{cos}x+c2\mathrm{sin}x\dots .\left(2\right)$
Putting $y=0$ for $x=0$ in equation (2), we get, $0=c1\mathrm{cos}0+c2sin0\to c1=0$
And putting $y=1$ for $x=1$ in equation (2), we get,
Putting the values of c1 and c2 in equation (2), we get, $y=\left(\frac{1}{\mathrm{sin}1}\right)\mathrm{sin}x$