# -7.1 transform u''+0.5u'+2u=0 into a system of first order eq

• MHB
• karush
In summary, to transform the given equation into a system of first order equations, we can use substitution by letting $u_1=u$ and $u_2=u'_1$. This yields the system of equations $u_1'=u_2$ and $u_2'=-0.5u_2-2u_1$. For the equation $u''+0.5u'+2u=3\sin t$, we can similarly define $u_1=u$ and $u_2=u'$ and obtain the system $u_1'=u_2$ and $u_2'=3\sin(t)-0.5u_2-2u_1$.
karush
Gold Member
MHB
transform the given equation into a system of first order equation$$u''+0.5u'+2u=0$$ok from examples it looks all we do is get rid of some of the primes and this is done by substitutionso if $u_1=u$ and $u_2=u'_1$
then $u_2=u'$ and $u'_2=u''$
then we have $u'_2+0.5u_2 +2u_1 = 0$then isolate $u'_2$ thus $u'_2=-0.5u_2-2u_1$ok the next problem is$$u''+0.5u'+2u=3\sin t$$ so ?

Last edited:
karush said:
transform the given equation into a system of first order equation$$u''+0.5u'+2u=0$$ok from examples it looks all we do is get rid of some of the primes and this is done by substitutionso if $u_1=u$ and $u_2=u'_1$
then $u_2=u'$ and $u'_2=u''$
then we have $u'_2+0.5u_2 +2u_1 = 0$then isolate $u'_2$ thus $u'_2=-0.5u_2-2u_1$
But you don't yet have a system of equations!
Your system of equations consist of the two equations
$u'_1= u_2$ and
$u'_2= -0.5u_2- 2u_1$.

ok the next problem is $$u''+0.5u'+2u=3\sin t$$ so ?
The same thing. Let $u_1= u$ and $u_2= u'$.
The given equation is $u_2'+ 0.5u_2+ 2u_1= 3\sin(t)$
which can be written $u_2'= 3\sin(t)- 0.5u_2- 2u_1$.

Your system of equations consist of the two equations
$u_1'= u_2$ and
$u_2'= 3\sin(t)- 0.5u_2- 2u_1$.

OK thank you much
I haven't been on the forum for a few days..

actually I plan on taking this class again in spring 2020 (UHM 307)
I just audited it last spring but really missed a lot of classes and homework

## 1. What is a first order equation?

A first order equation is an ordinary differential equation that contains only first derivatives of the dependent variable. It can be written in the form of dy/dx = f(x,y), where y is the dependent variable and x is the independent variable.

## 2. How do you transform a second order equation into a system of first order equations?

To transform a second order equation into a system of first order equations, we introduce a new variable, say u, and rewrite the equation as a system of two first order equations: u' = y and y' = f(x,y). This allows us to solve for both u and y simultaneously.

## 3. What is the purpose of transforming a second order equation into a system of first order equations?

Transforming a second order equation into a system of first order equations allows us to solve for both variables simultaneously and obtain a more general solution. It also allows us to use numerical methods to approximate the solution if an analytical solution is not possible.

## 4. What is the significance of the coefficients in the transformed system of first order equations?

The coefficients in the transformed system of first order equations represent the rate of change of the dependent variable with respect to the independent variable. They determine the behavior of the system and can provide insight into the dynamics of the original second order equation.

## 5. Are there any limitations to transforming a second order equation into a system of first order equations?

Yes, there are some limitations to this transformation. It may not be possible to transform every second order equation into a system of first order equations, especially if the equation is non-linear. In addition, the transformed system may be more complex and difficult to solve than the original equation.

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