91Ó°ÊÓ

The differential equation is,

$\mathrm{y}\text{'}\text{'}\left(\mathrm{θ}\right)+2\mathrm{y}\text{'}\left(\mathrm{θ}\right)+2\mathrm{y}\left(\mathrm{θ}\right)={\mathrm{e}}^{-\mathrm{θ}}\mathrm{³¦´Ç²õθ}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}…\left(1\right)$

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

$\mathrm{y}\text{'}\text{'}\left(\mathrm{θ}\right)+2\mathrm{y}\text{'}\left(\mathrm{θ}\right)+2\mathrm{y}\left(\mathrm{θ}\right)=0$

The auxiliary equation for the above equation,

${\mathrm{m}}^2+2\mathrm{m}+2=0$

Solve the auxiliary equation,

$\begin{array}{c}{\mathrm{m}}^2+2\mathrm{m}+2=0\\ \mathrm{m}=\frac{-2Â\pm \sqrt{4-8}}{2}\\ \mathrm{m}=\frac{-2Â\pm \sqrt{-4}}{2}\\ \mathrm{m}=-1Â\pm \mathrm{i}\end{array}$

The roots of the auxiliary equation are,

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

The complementary solution of the given equation is,

${\mathrm{y}}_{\mathrm{c}}={\mathrm{e}}^{-\mathrm{θ}}[{\mathrm{c}}_1\mathrm{³¦´Ç²õθ}+{\mathrm{c}}_2\mathrm{²õ¾\pm ²Ôθ}]$

According to the method of undetermined coefficients, to find a particular solution to the differential equation;

$\mathrm{ay}\text{'}\text{'}+\mathrm{by}\text{'}+\mathrm{cy}=\left\{\begin{array}{l}{\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{Î\pm ³Ù}}\mathrm{³¦´Ç²õβ³Ù}\\ \text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰}\mathrm{or}\\ {\mathrm{Ct}}^{\mathrm{m}}{\mathrm{e}}^{\mathrm{Î\pm ³Ù}}\mathrm{²õ¾\pm ²Ôβ³Ù}\end{array}\right\}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â€‰â¶Ä‰â€‰}......\left(2\right)$

For $\mathrm{β}≠0$, use the form

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^{\mathrm{s}}[({\mathrm{A}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{A}}_1\mathrm{t}+{\mathrm{A}}_0){\mathrm{e}}^{\mathrm{Î\pm ³Ù}}\mathrm{³¦´Ç²õβ³Ù}+{\mathrm{t}}^{\mathrm{s}}({\mathrm{B}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{B}}_1\mathrm{t}+{\mathrm{B}}_0){\mathrm{e}}^{\mathrm{Î\pm ³Ù}}\mathrm{²õ¾\pm ²Ôβ³Ù}]$

With s = 1 if$\mathrm{Î\pm }+\mathrm{¾\pm β}$ is a root of the associated auxiliary equation.

And s = 0 if $\mathrm{Î\pm }+\mathrm{¾\pm β}$is not a root of the associated auxiliary equation.

Comparing equations (1) and (2), we get;

M=0 and$\mathrm{Î\pm }=-1;\mathrm{β}=1$

Therefore,$\mathrm{Î\pm }+\mathrm{¾\pm β}=-1+\mathrm{i}$is a root of the associated auxiliary equation so here s =1.

Assume, the particular solution of equation (1),

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right)={\mathrm{t}}^{\mathrm{s}}({\mathrm{A}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{A}}_1\mathrm{t}+{\mathrm{A}}_0){\mathrm{e}}^{\mathrm{Î\pm ³\ae }}\mathrm{³¦´Ç²õβ³\ae }+{\mathrm{t}}^{\mathrm{s}}({\mathrm{B}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{B}}_1\mathrm{t}+{\mathrm{B}}_0){\mathrm{e}}^{\mathrm{Î\pm ³\ae }}\mathrm{²õ¾\pm ²Ôβ³\ae }\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)={\mathrm{θ}}^1\left({\mathrm{A}}_0\right){\mathrm{e}}^{(-1)\mathrm{θ}}\cos \left(1\right)\mathrm{θ}+{\mathrm{θ}}^1\left({\mathrm{B}}_0\right){\mathrm{e}}^{(-1)\mathrm{θ}}\sin \left(1\right)\mathrm{θ}\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)={\mathrm{θ\pm ð}}^{-\mathrm{θ}}({\mathrm{A}}_0\mathrm{³¦´Ç²õθ}+{\mathrm{B}}_0\mathrm{²õ¾\pm ²Ôθ})\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)={\mathrm{θ\pm ð}}^{-\mathrm{θ}}(\mathrm{A³¦´Ç²õθ}+\mathrm{B²õ¾\pm ²Ôθ})\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰}......\left(3\right)\end{array}$

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

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\text{'}\left(\mathrm{θ}\right)={\mathrm{e}}^{-\mathrm{θ}}[\mathrm{A}(-\mathrm{θ²õ¾\pm ²Ôθ}+\mathrm{³¦´Ç²õθ})+\mathrm{B}(\mathrm{賦´Ç²õθ}+\mathrm{²õ¾\pm ²Ôθ})]-{\mathrm{θ\pm ð}}^{-\mathrm{θ}}(\mathrm{A³¦´Ç²õθ}+\mathrm{B²õ¾\pm ²Ôθ})\\ {\mathrm{y}}_{\mathrm{p}}\text{'}\left(\mathrm{θ}\right)={\mathrm{e}}^{-\mathrm{θ}}[(-\mathrm{A}-\mathrm{B})\mathrm{θ²õ¾\pm ²Ôθ}+\mathrm{A³¦´Ç²õθ}+(\mathrm{B}-\mathrm{A})\mathrm{賦´Ç²õθ}+\mathrm{B²õ¾\pm ²Ôθ}]\\ {\mathrm{y}}_{\mathrm{p}}\text{'}\text{'}\left(\mathrm{θ}\right)={\mathrm{e}}^{-\mathrm{θ}}[\left(2\mathrm{A}\right)\mathrm{θ²õ¾\pm ²Ôθ}+(2\mathrm{B}-2\mathrm{A})\mathrm{³¦´Ç²õθ}+(-2\mathrm{B})\mathrm{賦´Ç²õθ}+(-2\mathrm{A}-2\mathrm{B})\mathrm{²õ¾\pm ²Ôθ}]\end{array}$

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

$\begin{array}{r}⇒\mathrm{y}\text{'}\text{'}\left(\mathrm{θ}\right)+2\mathrm{y}\text{'}\left(\mathrm{θ}\right)+2\mathrm{y}\left(\mathrm{θ}\right)={\mathrm{e}}^{-\mathrm{θ}}\mathrm{³¦´Ç²õθ}\\ ⇒{\mathrm{e}}^{-\mathrm{θ}}[\left(2\mathrm{A}\right)\mathrm{θ²õ¾\pm ²Ôθ}+(2\mathrm{B}-2\mathrm{A})\mathrm{³¦´Ç²õθ}+(-2\mathrm{B})\mathrm{賦´Ç²õθ}+(-2\mathrm{A}-2\mathrm{B})\mathrm{²õ¾\pm ²Ôθ}]\\ \text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}+2{\mathrm{e}}^{-\mathrm{θ}}[(-\mathrm{A}-\mathrm{B})\mathrm{θ²õ¾\pm ²Ôθ}+\mathrm{A³¦´Ç²õθ}+(\mathrm{B}-\mathrm{A})\mathrm{賦´Ç²õθ}+\mathrm{B²õ¾\pm ²Ôθ}]\\ +2{\mathrm{θ\pm ð}}^{-\mathrm{θ}}(\mathrm{A³¦´Ç²õθ}+\mathrm{B²õ¾\pm ²Ôθ})={\mathrm{e}}^{-\mathrm{θ}}\mathrm{³¦´Ç²õθ}\\ ⇒{\mathrm{e}}^{-\mathrm{θ}}[\left(2\mathrm{B}\right)\mathrm{³¦´Ç²õθ}+(-2\mathrm{A})\mathrm{²õ¾\pm ²Ôθ}]={\mathrm{e}}^{-\mathrm{θ}}\mathrm{³¦´Ç²õθ}\end{array}$

Comparing all coefficients of the above equation,

role="math" localid="1654949258619" $\begin{array}{rcl}-2\mathrm{A}& =& 0\text{ â¶Ä‰}⇒\mathrm{A}=0\\ 2\mathrm{B}& =& 1\text{ â¶Ä‰}⇒\mathrm{B}=\frac{1}{2}\end{array}$

Substitute the value of A and B in the equation (3),

role="math" localid="1654949410924"

Therefore, the particular solution of equation (1),

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right)=\frac{1}{2}{\mathrm{θ\pm ð}}^{-\mathrm{θ}}\mathrm{²õ¾\pm ²Ôθ}$

Therefore, the general solution is,

$$\begin{gathered} \mathrm{y}={\mathrm{y}}_{\mathrm{c}}\left(\mathrm{θ}\right)+{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{θ}\right) \\ \mathrm{y}={\mathrm{e}}^{-\mathrm{θ}}[{\mathrm{c}}_1\mathrm{³¦´Ç²õθ}+{\mathrm{c}}_2\mathrm{²õ¾\pm ²Ôθ}]+\frac{1}{2}{\mathrm{θ\pm ð}}^{-\mathrm{θ}}\mathrm{²õ¾\pm ²Ôθ} \end{gathered}$$