Question

# 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.

Second order linear equations

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

2021-05-19

We have to find the solution of the given initial value problem, $$y+y''=0, \ y(0)=0,\ y(1)=1$$
The differential equation can be written as, $$\displaystyle{\left({D}^{{2}}+{1}\right)}{y}={0}\ldots.{\left({1}\right)}$$
where $$\displaystyle{D}==\frac{{d}}{{\left.{d}{x}\right.}},{D}^{{2}}==\frac{{{d}^{{2}}}}{{{\left.{d}{x}\right.}^{{2}}}}$$
So the auxiliari equation of (1) is, $$D^2+1=0 \rightarrow D^2=-1 \rightarrow D=+-i$$
The required general solution is, $$\displaystyle{y}={c}{1}{\cos{{x}}}+{c}{2}{\sin{{x}}}\ldots.{\left({2}\right)}$$
Putting $$y=0$$ for $$x=0$$ in equation (2), we get, $$0=c1\cos0+c2sin0 \rightarrow c1=0$$
And putting $$y=1$$ for $$x=1$$ in equation (2), we get, $$1=c1\cos1+c2\sin1 \rightarrow c2=\frac{1}{\sin1} [as\ c1=0]$$
Putting the values of c1 and c2 in equation (2), we get, $$\displaystyle{y}={\left(\frac{{1}}{{\sin{{1}}}}\right)}{\sin{{x}}}$$