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### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# Find a general solution to the differential equation.${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}{\mathbf{+}}{\mathbf{6}}{\mathbf{y}}{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}{\mathbf{+}}{\mathbf{10}}{\mathbf{y}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}{\mathbf{=}}{\mathbf{10}}{{\mathbf{x}}}^{4}{\mathbf{+}}{\mathbf{24}}{{\mathbf{x}}}^{3}{\mathbf{+}}{\mathbf{2}}{{\mathbf{x}}}^{2}{\mathbf{-}}{\mathbf{12}}{\mathbf{x}}{\mathbf{+}}{\mathbf{18}}$

The general solution to the given differential equation is:

$\mathrm{y}={\mathrm{c}}_{1}{\mathrm{e}}^{-3\mathrm{x}}\mathrm{cosx}+{\mathrm{c}}_{2}{\mathrm{e}}^{-3\mathrm{x}}\mathrm{sinx}+{\mathrm{x}}^{4}-{\mathrm{x}}^{2}+2$.

See the step by step solution

## Step 1: Write the auxiliary equation of the given differential equation.

The differential equation is,

$\mathrm{y}\text{'}\text{'}\left(\mathrm{x}\right)+6\mathrm{y}\text{'}\left(\mathrm{x}\right)+10\mathrm{y}\left(\mathrm{x}\right)=10{\mathrm{x}}^{4}+24{\mathrm{x}}^{3}+2{\mathrm{x}}^{2}-12\mathrm{x}+18\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(1\right)$

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

$\mathrm{y}\text{'}\text{'}\left(\mathrm{x}\right)+6\mathrm{y}\text{'}\left(\mathrm{x}\right)+10\mathrm{y}\left(\mathrm{x}\right)=0$

The auxiliary equation for the above equation,

${\mathrm{m}}^{2}+6\mathrm{m}+10=0$

## Step 2: Find the complementary solution of the given equation.

Solve the auxiliary equation,

$\begin{array}{c}{\mathrm{m}}^{2}+6\mathrm{m}+10=0\\ \mathrm{m}=\frac{-6±\sqrt{36-40}}{2}\\ \mathrm{m}=\frac{-6±\sqrt{4}}{2}\\ \mathrm{m}=-3±\mathrm{i}\end{array}$

The roots of the auxiliary equation are,

${\mathrm{m}}_{1}=-3+\mathrm{i},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}_{2}=-3-\mathrm{i}$

The complementary solution of the given equation is,

${\mathrm{y}}_{\mathrm{c}}={\mathrm{c}}_{1}{\mathrm{e}}^{-3\mathrm{x}}\mathrm{cosx}+{\mathrm{c}}_{2}{\mathrm{e}}^{-3\mathrm{x}}\mathrm{sinx}$

## Step 3: Now find the particular solution to find a general solution for the equation.

Assume, the particular solution of equation (1),

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right)={\mathrm{Ax}}^{4}+{\mathrm{Bx}}^{3}+{\mathrm{Cx}}^{2}+\mathrm{Dx}+\mathrm{E}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(2\right)$

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

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\text{'}\left(\mathrm{t}\right)=4{\mathrm{Ax}}^{3}+3{\mathrm{Bx}}^{2}+2\mathrm{Cx}+\mathrm{D}\\ {\mathrm{y}}_{\mathrm{p}}\text{'}\text{'}\left(\mathrm{t}\right)=12{\mathrm{Ax}}^{2}+6\mathrm{Bx}+2\mathrm{C}\end{array}$

Substitute the value of ${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right),\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{y}}_{\mathrm{p}}\text{'}\left(\mathrm{t}\right)$ and ${\mathrm{y}}_{\mathrm{p}}\text{'}\text{'}\left(\mathrm{t}\right)$ the equation (1),

$\begin{array}{l}⇒\mathrm{y}\text{'}\text{'}\left(\mathrm{x}\right)+6\mathrm{y}\text{'}\left(\mathrm{x}\right)+10\mathrm{y}\left(\mathrm{x}\right)=10{\mathrm{x}}^{4}+24{\mathrm{x}}^{3}+2{\mathrm{x}}^{2}-12\mathrm{x}+18\\ ⇒12{\mathrm{Ax}}^{2}+6\mathrm{Bx}+2\mathrm{C}+6\left(4{\mathrm{Ax}}^{3}+3{\mathrm{Bx}}^{2}+2\mathrm{Cx}+\mathrm{D}\right)+10\left({\mathrm{Ax}}^{4}+{\mathrm{Bx}}^{3}+{\mathrm{Cx}}^{2}+\mathrm{Dx}+\mathrm{E}\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=10{\mathrm{x}}^{4}+24{\mathrm{x}}^{3}+2{\mathrm{x}}^{2}-12\mathrm{x}+18\\ ⇒10{\mathrm{Ax}}^{4}+\left(10\mathrm{B}+24\mathrm{A}\right){\mathrm{x}}^{3}+\left(12\mathrm{A}+18\mathrm{B}+10\mathrm{C}\right){\mathrm{x}}^{2}+\left(6\mathrm{B}+12\mathrm{C}+10\mathrm{D}\right)\mathrm{x}+\left(2\mathrm{C}+6\mathrm{D}+10\mathrm{E}\right)\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=10{\mathrm{x}}^{4}+24{\mathrm{x}}^{3}+2{\mathrm{x}}^{2}-12\mathrm{x}+18\end{array}$

Comparing all coefficients of the above equation,

$\begin{array}{c}10\mathrm{A}=10\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⇒\mathrm{A}=1\\ 10\mathrm{B}+24\mathrm{A}=24\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(3\right)\\ 12\mathrm{A}+18\mathrm{B}+10\mathrm{C}=2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(4\right)\\ 6\mathrm{B}+12\mathrm{C}+10\mathrm{D}=-12\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(5\right)\\ 2\mathrm{C}+6\mathrm{D}+10\mathrm{E}=18\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\dots \left(6\right)\end{array}$

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

$\begin{array}{c}10\mathrm{B}+24\left(1\right)=24\\ \mathrm{B}=0\end{array}$

Substitute the value of A and B in the equation (4),

$\begin{array}{c}12\left(1\right)+18\left(0\right)+10\mathrm{C}=2\\ \mathrm{C}=-1\end{array}$

Substitute the value of C and B in the equation (5),

$\begin{array}{c}6\left(0\right)+12\left(-1\right)+10\mathrm{D}=-12\\ \mathrm{D}=0\end{array}$

Substitute the value of C and D in the equation (6),

$\begin{array}{c}2\left(-1\right)+6\left(0\right)+10\mathrm{E}=18\\ \mathrm{E}=2\end{array}$

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

$\begin{array}{c}{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right)={\mathrm{Ax}}^{4}+{\mathrm{Bx}}^{3}+{\mathrm{Cx}}^{2}+\mathrm{Dx}+\mathrm{E}\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right)=\left(1\right){\mathrm{x}}^{4}+\left(0\right){\mathrm{x}}^{3}+\left(-1\right){\mathrm{x}}^{2}+\left(0\right)\mathrm{x}+\left(2\right)\\ {\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right)={\mathrm{x}}^{4}-{\mathrm{x}}^{2}+2\end{array}$

Therefore, the particular solution of equation (1),

${\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right)={\mathrm{x}}^{4}-{\mathrm{x}}^{2}+2$

## Step 4: Conclusion

Therefore, the general solution is,

$\begin{array}{c}\mathrm{y}={\mathrm{y}}_{\mathrm{c}}\left(\mathrm{t}\right)+{\mathrm{y}}_{\mathrm{p}}\left(\mathrm{t}\right)\\ \mathrm{y}={\mathrm{c}}_{1}{\mathrm{e}}^{-3\mathrm{x}}\mathrm{cosx}+{\mathrm{c}}_{2}{\mathrm{e}}^{-3\mathrm{x}}\mathrm{sinx}+{\mathrm{x}}^{4}-{\mathrm{x}}^{2}+2\end{array}$