Simultaneous differential equation

[suvadip]

Solve:
dx/dt +2y+e^x=t²
dy/dt-x+xe^x=0

Please help for this simultaneous differential equations

 

[Jester]

Re: simultaneous differential equation

Can we ask where this system came from?

 

[suvadip]

Re: simultaneous differential equation

Can we ask where this system came from?

This is purely a mathematical problem. It was set in a university exam. I dont have any further details about it.

 

[chisigma]

Re: simultaneous differential equation

Solve:
dx/dt +2y+e^x=t²
dy/dt-x+xe^x=0

Please help for this simultaneous differential equations

From the first equation...


$\displaystyle y= \frac{t^{2} - x^{\ '} - e^{x}}{2} \implies y^{\ '}= t - \frac{x^{\ ''}}{2} - \frac{x^{\ '}\ e^{x}}{2}$ (1)

... and inserting (1) into the second equation...

$\displaystyle x^{\ ''} + x^{\ '} e^{x} + 2 x (1-e^{x}) = 2 t$ (2)

... that is a second order ODE the solution of which will be done in a successive post...

Kind regards

$\chi$ $\sigma$

 

[Jester]

Re: simultaneous differential equation

This is purely a mathematical problem. It was set in a university exam. I dont have any further details about it.

The reason I ask - if it's from an exam and it says solve, then it has a solution.

 

[HallsofIvy]

Re: simultaneous differential equation

From the first equation...


$\displaystyle y= \frac{t^{2} - x^{\ '} - e^{x}}{2} \implies y^{\ '}= t - \frac{x^{\ ''}}{2} - \frac{x^{\ '}\ e^{x}}{2}$ (1)

... and inserting (1) into the second equation...

$\displaystyle x^{\ ''} + x^{\ '} e^{x} + 2 x (1-e^{x}) = 2 t$ (2)

... that is a second order ODE the solution of which will be done in a successive post...

Kind regards
$\chi$ $\sigma$

I look forward to seeing it! That looks to me like a very difficult non-linear equation.

 

[Klaas van Aarsen]

Re: simultaneous differential equation

I look forward to seeing it! That looks to me like a very difficult non-linear equation.

I just checked and found that W|A couldn't solve it.
Perhaps its solution didn't fit in the margin next to $\chi$ $\sigma$.

 

[chisigma]

Re: simultaneous differential equation

From the first equation...


$\displaystyle y= \frac{t^{2} - x^{\ '} - e^{x}}{2} \implies y^{\ '}= t - \frac{x^{\ ''}}{2} - \frac{x^{\ '}\ e^{x}}{2}$ (1)

... and inserting (1) into the second equation...

$\displaystyle x^{\ ''} + x^{\ '} e^{x} + 2 x (1-e^{x}) = 2 t$ (2)

... that is a second order ODE the solution of which will be done in a successive post...

The only chance to solve second order ODE we are arrived...

$\displaystyle x^{\ ''} + x^{\ '} e^{x} + 2 x (1-e^{x}) = 2 t$ (1)

... is the elimination of one of the term $x(t)$ or $x^{\ '}(t)$ and that fortunately can be done with the substitution $x (t)= - \ln \lambda (t)$, and doing that we obtain the new ODE...

$\displaystyle \lambda^{\ ''} =- 2\ \{t\ \lambda + \ln \lambda (\lambda -1)\} $ (2)

Using the identity...

$\displaystyle \lambda^{\ ''} = \frac{d \lambda^{\ '}}{d t} = \frac{d \lambda^{\ '}}{d \lambda}\ \frac{d\ \lambda}{d t} = \lambda^{\ '}\ \frac{d \lambda^{\ '}}{d \lambda}$ (3)

... we can write the (2) as...

$\displaystyle \lambda^{\ '}\ \frac{d \lambda^{\ '}}{d \lambda} =- 2\ \{t\ \lambda + \ln \lambda (\lambda -1)\} $ (4)

... and integrating (4) we obtain...

$\displaystyle \frac{\lambda^{\ '\ 2}}{2} = - t\ \lambda^{2} + \frac{1}{2}\ \lambda\ \{\lambda -2\ (\lambda-2)\ \ln \lambda -4\} + c_{1} \implies \lambda^{\ '} = \pm \sqrt {\lambda\ \{\lambda -2\ (\lambda-2)\ \ln \lambda -4\} - 2\ t\ \lambda^{2} + c_{1}}$ (5)

Writing (5) the original problem is 'half solved'... the feasibility of a complete solution will be analized is a successive post...

Kind regards

$\chi$ $\sigma$

 

[Jester]

Re: simultaneous differential equation

The only chance to solve second order ODE we are arrived...

$\displaystyle x^{\ ''} + x^{\ '} e^{x} + 2 x (1-e^{x}) = 2 t$ (1)

... is the elimination of one of the term $x(t)$ or $x^{\ '}(t)$ and that fortunately can be done with the substitution $x (t)= - \ln \lambda (t)$, and doing that we obtain the new ODE...

$\displaystyle \lambda^{\ ''} =- 2\ \{t\ \lambda + \ln \lambda (\lambda -1)\} $ (2)

Using the identity...

$\displaystyle \lambda^{\ ''} = \frac{d \lambda^{\ '}}{d t} = \frac{d \lambda^{\ '}}{d \lambda}\ \frac{d\ \lambda}{d t} = \lambda^{\ '}\ \frac{d \lambda^{\ '}}{d \lambda}$ (3)

... we can write the (2) as...

$\displaystyle \lambda^{\ '}\ \frac{d \lambda^{\ '}}{d \lambda} =- 2\ \{t\ \lambda + \ln \lambda (\lambda -1)\} $ (4)

... and integrating (4) we obtain...

$\displaystyle \frac{\lambda^{\ '\ 2}}{2} = - t\ \lambda^{2} + \frac{1}{2}\ \lambda\ \{\lambda -2\ (\lambda-2)\ \ln \lambda -4\} + c_{1} \implies \lambda^{\ '} = \pm \sqrt {\lambda\ \{\lambda -2\ (\lambda-2)\ \ln \lambda -4\} - 2\ t\ \lambda^{2} + c_{1}}$ (5)

Writing (5) the original problem is 'half solved'... the feasibility of a complete solution will be analized is a successive post...

Kind regards

$\chi$ $\sigma$

First, I don't believe that using $x = - \ln \lambda$ that (1) becomes (2). Second, even if it did, you simply cannot integrate like this! $t$ is not constant but the independent variable.

 

[chisigma]

Re: simultaneous differential equation

First, I don't believe that using $x = - \ln \lambda$ that (1) becomes (2). Second, even if it did, you simply cannot integrate like this! $t$ is not constant but the independent variable.

First I'll try to describe 'step by step' how (1) becomes (2). Starting from...


$\displaystyle x^{\ ''} + x^{\ '}\ e^{x} + 2\ x\ (1-e^{x})=2\ t$ (1)

... we set...

$\displaystyle x= - \ln \lambda \implies e^{x}= \frac{1}{\lambda} \implies x^{\ '}= - \frac{\lambda^{\ '}}{\lambda} \implies x^{\ ''}= \frac{\lambda^{\ '}}{\lambda^{2}} - \frac{\lambda^{\ ''}}{\lambda} $ (2)

... and from (1) and (2)...

$\displaystyle \frac{\lambda^{\ '}}{\lambda^{2}} - \frac{\lambda^{\ ''}}{\lambda} - \frac{\lambda^{\ '}}{\lambda^{2}} - 2\ \ln \lambda\ (1-\frac{1}{\lambda}) = 2\ t \implies \lambda^{\ ''} + 2\ \{t\ \lambda + \ln \lambda\ (\lambda-1)\}=0$ (3)

Kind regards

$\chi$ $\sigma$

 

[Jester]

I believe that if

$x = - \ln \lambda$ then $x' = - \dfrac{\lambda'}{\lambda}$ - correct

but

$x'' = - \dfrac{\lambda ''}{\lambda} + \dfrac{\lambda'^2}{\lambda^2}$.