Question

Solve the following differential equations by the methods discussed above and compare computer solutions.

$4y^{\text{'}\text{'}}+12y^{\text{'}}+9=0$

Short answer

The solution is $y=(Ax+B)e^{-\frac{3}{2}x}$.

Step-by-step solution

Given information            

A differential equation is given as $4y"+y\text{'}+9=0$.

Differential equation. 

A differential equation of the form $(D-a)(D-b)y=0$has general solution

$y=(Ax+B)e^{ax}.$

Find the solution of given differential equation.

A differential equation is given as$4y"\text{'}+y\text{'}+9=0$.

Suppose that $D=\frac{d}{dx}$.

Then

$$\begin{gathered} Dy=\frac{dy}{dx} \\ Dy=y^{\text{'}} \\ {D}^{2}y=\frac{d}{dx}\left(\frac{dy}{dx}\right) \\ \frac{d^{2}y}{d{x}^2}=y^{\text{'}\text{'}} \end{gathered}$$

Given differential equation is$4y"+y\text{'}+9=0$.

Above equation becomes,

$$\begin{gathered} 4D^{2}y+12Dy+9=0 \\ (2D+3)(2D+3)y=0 \\ (2D+3)y=0\text{and}(2D+3)y=0 \\ \left(D+\frac{3}{2}\right)y=0\text{and}\left(D+\frac{3}{2}\right)y=0 \end{gathered}$$

Solve the problem further and get

$\Rightarrow \left[D-\left(-\frac{3}{2}\right)\right]y=0\text{and}\left[D-\left(-\frac{3}{2}\right)\right]y=0$

The general solution will be given by $y=(Ax+B)e^{-\frac{3}{2}x}$.