Solutions of DE System & 2nd Order Differential Equation

In summary, the two solutions to the differential equation system are different because in the second case, there is a minor arithmetic mistake in the solution.
  • #1
WMDhamnekar
MHB
378
28
Hello,
$\vec{x'}=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$

Now i got the solution to this differential equation system as

$\vec{x}(t)=c_1e^{-t}\small\begin{pmatrix}-1\\1\end{pmatrix}$+$c_2e^{4t}\small\begin{pmatrix}2\\3\end{pmatrix}$+$t\small\begin{pmatrix}3\\\frac{-5}{2}\end{pmatrix}$+$\small\begin{pmatrix}-2.75\\2.875\end{pmatrix}$

Now i converted this differential equation system into ordinary differential equation $y''-3y'-4y+12t-2=0$

I got solution to this DE as $y=C_1e^{-x}+C_2e^{4x}+3t-\frac12$.

Now my question why there is diference in these two solutions?
 
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  • #2
Dhamnekar Winod said:
Hello,
$\vec{x'}=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$

Now i got the solution to this differential equation system as

$\vec{x}(t)=c_1e^{-t}\small\begin{pmatrix}-1\\1\end{pmatrix}$+$c_2e^{4t}\small\begin{pmatrix}2\\3\end{pmatrix}$+$t\small\begin{pmatrix}3\\\frac{-5}{2}\end{pmatrix}$+$\small\begin{pmatrix}-2.75\\2.875\end{pmatrix}$

Now i converted this differential equation system into ordinary differential equation $y''-3y'-4y+12t-2=0$

I got solution to this DE as $y=C_1e^{-x}+C_2e^{4x}+3t-\frac12$.

Now my question why there is diference in these two solutions?

Hi Dhamnekar Winod, welcome to MHB! ;)

Your solution contains $x$ on the right hand side. I presume it should be $t$?
Either way, if I feed the latter equation to Wolfram|Alpha, I get:
$$y(t)=c_1e^{-t}+c_2e^{4t}+3t-\frac{11}{4}$$
So:
  • It seems to be the solution for $x(t)$ rather than $y(t)$.
  • Your equation is correct, although for $x(t)$ rather than $y(t)$.
  • There is a minor arithmetic mistake somewhere in your solution.
 
  • #3
Hello, The derivative of $x_1=C_1e^{-t}+C_23e^{4t}-2.5t+2.875$in the differential equation system.But in the second case,it is different Why?
 

FAQ: Solutions of DE System & 2nd Order Differential Equation

1. What is the difference between a solution of a DE system and a 2nd order differential equation?

A solution of a DE system is a set of equations that satisfy a system of differential equations, while a solution of a 2nd order differential equation is an equation that satisfies a single 2nd order differential equation.

2. Can a solution of a DE system also be a solution of a 2nd order differential equation?

Yes, it is possible for a solution of a DE system to also satisfy a 2nd order differential equation. However, not all solutions of a DE system will also satisfy a 2nd order differential equation.

3. How do you find the solution of a DE system?

The solution of a DE system can be found by solving the system of equations using various methods such as substitution, elimination, or matrix operations. Some systems may also have analytical solutions, while others may require numerical methods.

4. What are initial conditions and why are they important in finding solutions of DE systems and 2nd order differential equations?

Initial conditions refer to the values of the variables at a specific point in time or position. They are important because they help determine the unique solution to a DE system or 2nd order differential equation. Without initial conditions, there may be multiple solutions that satisfy the equations.

5. Can a DE system or 2nd order differential equation have more than one solution?

Yes, a DE system or 2nd order differential equation can have multiple solutions. In fact, most DE systems and 2nd order differential equations have an infinite number of solutions. However, some may have a unique solution depending on the initial conditions.

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