Collin's question via email about solving a DE using Laplace Transforms

In summary, the given differential equation is solved by taking the Laplace transform and using partial fractions. The solution is then found using the inverse Laplace transform, resulting in the equation y = H(t-7) - 1/8sin[2(t-7)]H(t-7) - 5sin(2t). The solution can also be found using residue calculus.
  • #1
Prove It
Gold Member
MHB
1,465
24
Solve the following Differential Equation:

$\displaystyle \begin{align*} y'' + 4\,y = \mathrm{H}\,\left( t - 7 \right) \textrm{ with } y\left( 0 \right) = 0 \textrm{ and } y'\left( 0 \right) = -10 \end{align*}$

Taking the Laplace Transform of both sides we have

$\displaystyle \begin{align*} \mathcal{L}\,\left\{ y'' + 4\,y \right\} &= \mathcal{L}\,\left\{ \mathrm{H}\,\left( t - 7 \right) \right\} \\ s^2\,Y\left( s \right) - s\,y\left( 0 \right) - y'\left( 0 \right) + 4\,Y\left( s \right) &= \frac{\mathrm{e}^{-7\,s}}{s} \\ s^2\,Y\left( s \right) + 10 + 4\,Y\left( s \right) &= \frac{ \mathrm{e}^{-7\,s}}{s} \\ \left( s^2 + 4 \right) \, Y\left( s \right) &= \frac{\mathrm{e}^{-7\,s}}{s} - 10 \\ Y\left( s \right) &= \frac{\mathrm{e}^{-7\,s}}{s\,\left( s^2 + 4 \right) } - \frac{10}{s^2 + 4} \end{align*}$

So to solve the DE, all that we require now is to take the Inverse Laplace Transform of this result. To do this with the first term, we will need to use Partial Fractions:

$\displaystyle \begin{align*} \frac{A}{s} + \frac{B\,s + C}{s^2 + 4} &\equiv \frac{1}{s\,\left( s^2 + 4 \right) } \\ A\,\left( s^2 + 4 \right) + \left( B\,s + C \right) \, s &= 1 \end{align*}$

Let $\displaystyle \begin{align*} s = 0 \end{align*}$ to find $\displaystyle \begin{align*} 4\,A = 1 \implies A = \frac{1}{4} \end{align*}$.

The coefficient of $\displaystyle \begin{align*} s^2 \end{align*}$ is $\displaystyle \begin{align*} A + B \end{align*}$ on the left and $\displaystyle \begin{align*} 0 \end{align*}$ on the right, and as $\displaystyle \begin{align*} A = \frac{1}{4} \end{align*}$ we have $\displaystyle \begin{align*} \frac{1}{4} + B = 0 \implies B = -\frac{1}{4} \end{align*}$.

The coefficient of $\displaystyle \begin{align*} s \end{align*}$ is $\displaystyle \begin{align*} C \end{align*}$ on the left and $\displaystyle \begin{align*} 0 \end{align*}$ on the right, thus $\displaystyle \begin{align*} C = 0 \end{align*}$.

Therefore $\displaystyle \begin{align*} \frac{1}{s\,\left( s^2 + 4 \right) } = \frac{1}{4\,s} - \frac{s}{4\,\left( s^2 + 4 \right) } \end{align*}$. So that means

$\displaystyle \begin{align*} y &= \mathcal{L}^{-1}\,\left\{ \frac{\mathrm{e}^{-7\,s}}{s\,\left( s^2 + 4 \right) } - \frac{10}{s^2 + 4} \right\} \\ &= \frac{1}{4} \mathcal{L}^{-1}\,\left\{ \frac{\mathrm{e}^{-7\,s}}{s} \right\} - \frac{1}{8} \,\mathcal{L}^{-1}\,\left\{ \mathrm{e}^{-7\,s} \,\left( \frac{2}{ s^2 + 2^2 } \right) \right\} - 5\,\mathcal{L}^{-1}\,\left\{ \frac{2}{s^2 + 2^2} \right\} \end{align*}$

The first and third terms are very straightforward. For the second term, we need to make use of the rule $\displaystyle \begin{align*} \mathcal{L}\,\left\{ f\left( t - c \right) \, \mathrm{H}\,\left( t - c \right) \right\} = \mathrm{e}^{-c\,s}\,F\left( s \right) \end{align*}$. We can read off that $\displaystyle \begin{align*} F\left( s \right) = \frac{2}{ s^2 + 2^2} \end{align*}$ and thus $\displaystyle \begin{align*} f\left( t \right) = \sin{ \left( 2\,t \right) } \end{align*}$. From here we can get that $\displaystyle \begin{align*} f\left( t - 7 \right) = \sin{ \left[ 2\,\left( t - 7 \right) \right] } \end{align*}$, and finally the solution to the DE is

$\displaystyle \begin{align*} y = \mathrm{H}\,\left( t - 7 \right) - \frac{1}{8}\sin{ \left[ 2\,\left( t - 7 \right) \right] }\,\mathrm{H}\,\left( t - 7 \right) - 5 \sin{ \left( 2\,t \right) } \end{align*}$.
 
Mathematics news on Phys.org
  • #2
Careful students will check the solution to ensure that it actually is a solution to the DE.
 
  • #3
By using the inversion formula and residue calculus I get [tex]\begin{split}
y(t) &= 10 \left( \frac{e^{2it}}{4i} + \frac{e^{-2it}}{-4i}\right) + \left( \frac14 + \frac{e^{2i(t-7)}}{(2i)(4i)}
+ \frac{e^{-2i(t-7)}}{(-2i)(-4i)} \right)H(t-7) \\
&= 5\sin(2t) + \left(\tfrac14 - \tfrac14\cos(2(t-7))\right)H(t-7).\end{split}[/tex] I feel this involved less work, with less scope for error, than manipulating [itex]\dfrac{10}{s^2 + 4} + \dfrac{e^{-7s}}{s(s^2+4)}[/itex] into a form where its inverse transform can be read from tables.
 

FAQ: Collin's question via email about solving a DE using Laplace Transforms

1) What is a differential equation (DE)?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many real-world phenomena in physics, engineering, and other scientific fields.

2) How do Laplace Transforms relate to solving DEs?

Laplace Transforms are a mathematical tool used to solve differential equations. They transform a differential equation into an algebraic equation, which can be easier to solve. This method is particularly useful for solving DEs with complex or non-constant coefficients.

3) Can all DEs be solved using Laplace Transforms?

No, not all DEs can be solved using Laplace Transforms. Some DEs are not suitable for this method, or they may require advanced techniques. However, Laplace Transforms are a powerful tool that can be applied to many types of differential equations.

4) What are the advantages of using Laplace Transforms to solve DEs?

One of the main advantages of using Laplace Transforms is that they can simplify the process of solving complex differential equations. They also allow for the use of algebraic techniques, which can be more familiar and easier to work with compared to the traditional methods of solving DEs.

5) Are there any limitations to using Laplace Transforms for solving DEs?

While Laplace Transforms can be a useful tool, they have some limitations. They may not be suitable for certain types of DEs, and they can be more challenging to use for solving boundary value problems. Additionally, some equations may require additional techniques or approximations to obtain a solution.

Similar threads

Replies
2
Views
10K
Replies
1
Views
10K
Replies
4
Views
10K
Replies
1
Views
10K
Replies
1
Views
9K
Replies
1
Views
10K
Replies
1
Views
10K
Replies
1
Views
9K
Replies
1
Views
10K
Back
Top