91Ó°ÊÓ

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

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

For,$\mathbf{β}≠\mathbf{0}$ 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.

The differential equation is,

$\mathrm{y}\text{'}\text{'}-2\mathrm{y}\text{'}+\mathrm{y}=7{\mathrm{e}}^{\mathrm{t}}\mathrm{cost}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰}.....\left(1\right)$

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

$\mathrm{y}\text{'}\text{'}-2\mathrm{y}\text{'}+\mathrm{y}=0$

The auxiliary equation for the above equation,

${\mathrm{m}}^2-2\mathrm{m}+1=0$

Solve the auxiliary equation,

$\begin{array}{c}{\mathrm{m}}^2-2\mathrm{m}+1=0\\ {(\mathrm{m}-1)}^2=0\end{array}$

The roots of the auxiliary equation are,

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

The complementary solution of the given equation is,

${\mathrm{y}}_{\mathrm{c}}={\mathrm{c}}_1{\mathrm{e}}^{\mathrm{t}}+{\mathrm{c}}_2{\mathrm{te}}^{\mathrm{t}}$

To find a particular solution to the differential equation

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

Compare with the given differential equation,

$\mathrm{y}\text{'}\text{'}-2\mathrm{y}\text{'}+\mathrm{y}=7{\mathrm{e}}^{\mathrm{t}}\mathrm{cost}$

We have,

$\mathrm{m}=0,\text{ }\mathrm{Î\pm }=1,\text{ â¶Ä‰}\mathrm{β}=1$

And

$\mathrm{Î\pm }+\mathrm{¾\pm β}=1+\mathrm{i}≠{\mathrm{m}}_1={\mathrm{m}}_2$

Therefore, we get

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

The particular solution to the differential equation for m = 0,

$\begin{array}{l}{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^{\mathrm{s}}[({\mathrm{A}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{A}}_1\mathrm{t}+{\mathrm{A}}_0){\mathrm{e}}^{\mathrm{Î\pm ³Ù}}\mathrm{³¦´Ç²õβ³Ù}+{\mathrm{t}}^{\mathrm{s}}({\mathrm{B}}_{\mathrm{m}}{\mathrm{t}}^{\mathrm{m}}+...+{\mathrm{B}}_1\mathrm{t}+{\mathrm{B}}_0){\mathrm{e}}^{\mathrm{Î\pm ³Ù}}\mathrm{²õ¾\pm ²Ôβ³Ù}]\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{t}}^0[\left({\mathrm{A}}_0\right){\mathrm{e}}^{\left(1\right)\mathrm{t}}\cos \left(1\right)\mathrm{t}+{\mathrm{t}}^0\left({\mathrm{B}}_0\right){\mathrm{e}}^{\left(1\right)\mathrm{t}}\sin \left(1\right)\mathrm{t}]\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{x}\right)={\mathrm{A}}_0{\mathrm{e}}^{\mathrm{t}}\mathrm{cost}+{\mathrm{B}}_0{\mathrm{e}}^{\mathrm{t}}\mathrm{sint}\end{array}$