Q2RP - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Find a general solution to the given differential equation.$\mathbf{49}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{14}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$

Step-by-Step Solution

Answer

$\mathbf{y}\mathbf{=}{\mathbf{c}}_1{\mathbf{e}}^{\frac{-1}{7}\mathbf{t}}\mathbf{+}{\mathbf{c}}_2{\mathbf{te}}^{\frac{-1}{7}\mathbf{t}}$

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

The given differential equation is$\mathbf{49}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{14}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}.$

 

The auxiliary equation for the above equation$\mathbf{49}{\mathbf{m}}^2\mathbf{+}\mathbf{14}\mathbf{m}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{0}.$

 

Step 2: Find the roots of the auxiliary equation

Solve the auxiliary equation,

 $\begin{array}{c}\mathbf{49}{\mathbf{m}}^2\mathbf{+}\mathbf{14}\mathbf{m}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{0}\\ \mathbf{49}{\mathbf{m}}^2\mathbf{+}\mathbf{7}\mathbf{m}\mathbf{+}\mathbf{7}\mathbf{m}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{0}\\ \mathbf{7}\mathbf{m}\left(7m+1\right)\mathbf{+}\mathbf{1}\left(7m+1\right)\mathbf{=}\mathbf{0}\\ \left(7m+1\right)\left(7m+1\right)\mathbf{=}\mathbf{0}\end{array}$

The roots of the auxiliary equation are${\mathbf{m}}_1\mathbf{=}\frac{-1}{7}\&{\mathbf{m}}_2\mathbf{=}\frac{-1}{7}.$

Thus, the general solution of the given equation is$\mathbf{y}\mathbf{=}{\mathbf{c}}_1{\mathbf{e}}^{\frac{-1}{7}\mathbf{t}}\mathbf{+}{\mathbf{c}}_2{\mathbf{te}}^{\frac{-1}{7}\mathbf{t}}.$