Answer: (a) \((\cosh x+\sinh x)^{\frac{1}{2}}=\mathrm{e}^{x/2}\).
(b) The solution satisfying the given conditions is
\(y=\mathrm{e}^{-x/2}\left(-\frac{1}{3}\cos\left(\frac{\sqrt{11}}{2}x\right)+\frac{1}{\sqrt{11}}\sin\left(\frac{\sqrt{11}}{2}x\right)\right)+\frac{4}{3}\mathrm{e}^{x/2}.\)
(a) Using \(\cosh x=\frac{\mathrm{e}^x+\mathrm{e}^{-x}}{2}\) and \(\sinh x=\frac{\mathrm{e}^x-\mathrm{e}^{-x}}{2}\),
\(\cosh x+\sinh x=\frac{\mathrm{e}^x+\mathrm{e}^{-x}}{2}+\frac{\mathrm{e}^x-\mathrm{e}^{-x}}{2}=\mathrm{e}^x.\)
Hence
\((\cosh x+\sinh x)^{1/2}=(\mathrm{e}^x)^{1/2}=\mathrm{e}^{x/2}.\)
(b) From part (a), the equation becomes
\(y''+y'+3y=5\mathrm{e}^{x/2}.\)
For the complementary function, solve
\(m^2+m+3=0.\)
This gives
\(m=\frac{-1\pm i\sqrt{11}}{2}.\)
So
\(y_c=\mathrm{e}^{-x/2}\left(A\cos\left(\frac{\sqrt{11}}{2}x\right)+B\sin\left(\frac{\sqrt{11}}{2}x\right)\right).\)
For a particular integral, try
\(y_p=k\mathrm{e}^{x/2}.\)
Then
\(y_p'=\frac12k\mathrm{e}^{x/2},\qquad y_p''=\frac14k\mathrm{e}^{x/2}.\)
Substitute into the differential equation:
\(\frac14k\mathrm{e}^{x/2}+\frac12k\mathrm{e}^{x/2}+3k\mathrm{e}^{x/2}=5\mathrm{e}^{x/2}.\)
So
\(\left(\frac14+\frac12+3\right)k=5 \Rightarrow \frac{15}{4}k=5,\)
hence
\(k=\frac43.\)
Therefore
\(y=\mathrm{e}^{-x/2}\left(A\cos\left(\frac{\sqrt{11}}{2}x\right)+B\sin\left(\frac{\sqrt{11}}{2}x\right)\right)+\frac43\mathrm{e}^{x/2}.\)
Use the conditions.
When \(x=0\), \(y=1\):
\(1=A+\frac43,\)
so
\(A=-\frac13.\)
Differentiate:
\(y'=\mathrm{e}^{-x/2}\left(-\frac{\sqrt{11}}{2}A\sin\left(\frac{\sqrt{11}}{2}x\right)+\frac{\sqrt{11}}{2}B\cos\left(\frac{\sqrt{11}}{2}x\right)\right)-\frac12\mathrm{e}^{-x/2}\left(A\cos\left(\frac{\sqrt{11}}{2}x\right)+B\sin\left(\frac{\sqrt{11}}{2}x\right)\right)+\frac23\mathrm{e}^{x/2}.\)
When \(x=0\), \(y'=\frac43\):
\(\frac43=\frac{\sqrt{11}}{2}B-\frac12A+\frac23.\)
Substitute \(A=-\frac13\):
\(\frac43=\frac{\sqrt{11}}{2}B+\frac16+\frac23=\frac{\sqrt{11}}{2}B+\frac56.\)
So
\(\frac12=\frac{\sqrt{11}}{2}B,\)
giving
\(B=\frac{1}{\sqrt{11}}.\)
Hence the required solution is
\(y=\mathrm{e}^{-x/2}\left(-\frac{1}{3}\cos\left(\frac{\sqrt{11}}{2}x\right)+\frac{1}{\sqrt{11}}\sin\left(\frac{\sqrt{11}}{2}x\right)\right)+\frac{4}{3}\mathrm{e}^{x/2}.\)
