91Ó°ÊÓ

The differential equation is,

$\mathrm{y}\text{'}\text{'}\left(\mathrm{θ}\right)-\mathrm{y}\left(\mathrm{θ}\right)=\mathrm{²õ¾\pm ²Ôθ}-{\mathrm{e}}^{2\mathrm{θ}}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰}......\left(1\right)$

Write the homogeneous differential equation of the equation (1),

$\mathrm{y}\text{'}\text{'}\left(\mathrm{θ}\right)-\mathrm{y}\left(\mathrm{θ}\right)=0$

The auxiliary equation for the above equation,

$\begin{array}{c}{\mathrm{m}}^2-1=0\\ \mathrm{m}=Â\pm 1\end{array}$

The root of an auxiliary equation is,

${\mathrm{m}}_1=1,\text{ â¶Ä‰â¶Ä‰}{\mathrm{m}}_2=-1$

The complementary solution of the given equation is,

${\mathrm{y}}_{\mathrm{c}}={\mathrm{c}}_1{\mathrm{e}}^{\mathrm{θ}}+{\mathrm{c}}_2{\mathrm{e}}^{-\mathrm{θ}}$

Assume, the particular solution of equation (1),

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)=\mathrm{A²õ¾\pm ²Ôθ}+\mathrm{\mu þ³¦´Ç²õθ}+{\mathrm{Ce}}^{2\mathrm{θ}}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}......\left(2\right)$

Now find the first and second derivatives of the above equation,

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\text{'}\left(\mathrm{θ}\right)=\mathrm{´¡³¦´Ç²õθ}-\mathrm{B²õ¾\pm ²Ôθ}+2{\mathrm{Ce}}^{2\mathrm{θ}}\\ {\mathrm{y}}_{\mathrm{p}}\text{'}\text{'}\left(\mathrm{θ}\right)=-\mathrm{A²õ¾\pm ²Ôθ}-\mathrm{\mu þ³¦´Ç²õθ}+4{\mathrm{Ce}}^{2\mathrm{θ}}\end{array}$

Substitute the value of ${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)$and ${\mathrm{y}}_{\mathrm{p}}\text{'}\text{'}\left(\mathrm{θ}\right)$the equation (1),

$\begin{array}{c}\mathrm{y}\text{'}\text{'}\left(\mathrm{θ}\right)-\mathrm{y}\left(\mathrm{θ}\right)=\mathrm{²õ¾\pm ²Ôθ}-{\mathrm{e}}^{2\mathrm{θ}}\\ -\mathrm{A²õ¾\pm ²Ôθ}-\mathrm{\mu þ³¦´Ç²õθ}+4{\mathrm{Ce}}^{2\mathrm{θ}}-[\mathrm{A²õ¾\pm ²Ôθ}+\mathrm{\mu þ³¦´Ç²õθ}+{\mathrm{Ce}}^{2\mathrm{θ}}]=\mathrm{²õ¾\pm ²Ôθ}-{\mathrm{e}}^{2\mathrm{θ}}\\ -2\mathrm{A²õ¾\pm ²Ôθ}-2\mathrm{\mu þ³¦´Ç²õθ}+3{\mathrm{Ce}}^{2\mathrm{θ}}=\mathrm{²õ¾\pm ²Ôθ}-{\mathrm{e}}^{2\mathrm{θ}}\end{array}$

Comparing all coefficients of the above equation,

role="math" localid="1655134375109" $\begin{array}{c}-2\mathrm{A}=1\text{ â¶Ä‰â¶Ä‰}⇒\mathrm{A}=\frac{-1}{2}\\ -2\mathrm{B}=0\text{ â¶Ä‰â¶Ä‰}⇒\mathrm{B}=0\\ 3\mathrm{C}=-1\text{ â¶Ä‰â¶Ä‰}⇒\mathrm{C}=\frac{-1}{3}\end{array}$

Substitute the value of A, B, and C in the equation (2),

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)=-\frac{1}{2}\mathrm{²õ¾\pm ²Ôθ}+\left(0\right)\mathrm{³¦´Ç²õθ}-\frac{1}{3}{\mathrm{e}}^{2\mathrm{θ}}\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)=-\frac{1}{2}\mathrm{²õ¾\pm ²Ôθ}-\frac{1}{3}{\mathrm{e}}^{2\mathrm{θ}}\end{array}$

Therefore, the general solution is,

$\begin{array}{c}\mathrm{y}={\mathrm{y}}_{\mathrm{c}}\left(\mathrm{θ}\right)+{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)\\ \mathrm{y}={\mathrm{c}}_1{\mathrm{e}}^{\mathrm{θ}}+{\mathrm{c}}_2{\mathrm{e}}^{-\mathrm{θ}}-\frac{1}{2}\mathrm{²õ¾\pm ²Ôθ}-\frac{1}{3}{\mathrm{e}}^{2\mathrm{θ}}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰}......\left(3\right)\end{array}$

Given the initial condition,

$\mathrm{y}\left(0\right)=1,\text{ â¶Ä‰â¶Ä‰â¶Ä‰}\mathrm{y}\text{'}\left(0\right)=-1$

Substitute the value of y = 1 and $\mathrm{θ}=0$in the equation (3),

role="math" localid="1655134708459" $\begin{array}{c}\mathrm{y}={\mathrm{c}}_1{\mathrm{e}}^{\mathrm{θ}}+{\mathrm{c}}_2{\mathrm{e}}^{-\mathrm{θ}}-\frac{1}{2}\mathrm{²õ¾\pm ²Ôθ}-\frac{1}{3}{\mathrm{e}}^{2\mathrm{θ}}\\ 1={\mathrm{c}}_1{\mathrm{e}}^0+{\mathrm{c}}_2{\mathrm{e}}^{-0}-\frac{1}{2}\sin \left(0\right)-\frac{1}{3}{\mathrm{e}}^{2\left(0\right)}\\ 1={\mathrm{c}}_1+{\mathrm{c}}_2-\frac{1}{3}\\ {\mathrm{c}}_1+{\mathrm{c}}_2=\frac{4}{3}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}......\left(4\right)\end{array}$

Now find the derivative of the equation (3),

role="math" localid="1655134750830" $\mathrm{y}\text{'}={\mathrm{c}}_1{\mathrm{e}}^{\mathrm{θ}}-{\mathrm{c}}_2{\mathrm{e}}^{-\mathrm{θ}}-\frac{1}{2}\mathrm{³¦´Ç²õθ}-\frac{2}{3}{\mathrm{e}}^{2\mathrm{θ}}$

Substitute the value of y’ = -1 and $\mathrm{θ}=0$in the above equation,

role="math" localid="1655134857938" $\begin{array}{c}\mathrm{y}\text{'}={\mathrm{c}}_1{\mathrm{e}}^{\mathrm{θ}}-{\mathrm{c}}_2{\mathrm{e}}^{-\mathrm{θ}}-\frac{1}{2}\mathrm{³¦´Ç²õθ}-\frac{2}{3}{\mathrm{e}}^{2\mathrm{θ}}\\ -1={\mathrm{c}}_1{\mathrm{e}}^0-{\mathrm{c}}_2{\mathrm{e}}^{-0}-\frac{1}{2}\cos \left(0\right)-\frac{2}{3}{\mathrm{e}}^{2\left(0\right)}\\ -1={\mathrm{c}}_1-{\mathrm{c}}_2-\frac{1}{2}-\frac{2}{3}\\ {\mathrm{c}}_1-{\mathrm{c}}_2=\frac{1}{6}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰}.....\left(5\right)\end{array}$

Solve the equation (4) and (5),

role="math" localid="1655134925741" $\begin{array}{l}{\mathrm{c}}_1+{\mathrm{c}}_2=\frac{4}{3}\\ {\mathrm{c}}_1-{\mathrm{c}}_2=\frac{1}{6}\\ \stackrel{¯}{\begin{array}{l}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â€‰}2{\mathrm{c}}_1=\frac{9}{6}\\ \text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}{\mathrm{c}}_1=\frac{3}{4}\end{array}}\end{array}$

Substitute the value of ${\mathrm{c}}_1$in the equation (4),

role="math" localid="1655135004141" $\begin{array}{c}{\mathrm{c}}_1+{\mathrm{c}}_2=\frac{4}{3}\\ \frac{3}{4}+{\mathrm{c}}_2=\frac{4}{3}\\ {\mathrm{c}}_2=\frac{7}{12}\end{array}$

Substitute the value of ${\mathrm{c}}_1$and ${\mathrm{c}}_2$in the equation (3),

role="math" localid="1655135133890" $\begin{array}{c}\mathrm{y}={\mathrm{c}}_1{\mathrm{e}}^{\mathrm{θ}}+{\mathrm{c}}_2{\mathrm{e}}^{-\mathrm{θ}}-\frac{1}{2}\mathrm{²õ¾\pm ²Ôθ}-\frac{1}{3}{\mathrm{e}}^{2\mathrm{θ}}\\ \mathrm{y}=\frac{3}{4}{\mathrm{e}}^{\mathrm{θ}}+\frac{7}{12}{\mathrm{e}}^{-\mathrm{θ}}-\frac{1}{2}\mathrm{²õ¾\pm ²Ôθ}-\frac{1}{3}{\mathrm{e}}^{2\mathrm{θ}}\end{array}$

Thus, the solution to the initial value is:

$\mathrm{y}=\frac{3}{4}{\mathrm{e}}^{\mathrm{θ}}+\frac{7}{12}{\mathrm{e}}^{-\mathrm{θ}}-\frac{1}{2}\mathrm{²õ¾\pm ²Ôθ}-\frac{1}{3}{\mathrm{e}}^{2\mathrm{θ}}$