Finding the explicit solution to the IVP

In summary, the conversation discusses finding the explicit solution to the initial value problem $xdx + ye^{-x}dy=0$, $y(0) =1$. The process involves manipulating the equation to obtain an implicit solution, then separating the variables and integrating to obtain the explicit solution. Both participants in the conversation arrive at the same explicit solution, $y= \sqrt{2e^x-2xe^x-1}$.
  • #1
shamieh
539
0
Find the explicit solution to the IVP.

$xdx + ye^{-x}dy=0$, $y(0) =1$
so I did some manipulation to get
$ye^{-x}dy= -xdx$ ==> $\frac{dy}{dx}=\frac{-x}{ye^{-x}}$

but now I'm confused on what to do. What I found above is the implicit solution right? So do I just need to get $y'$ on the left side by multiplying through with a $dx$ and then just plug a $0$ in for $x$ and a $1$ in for $y$ to get the explicit solution??
 
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  • #2
An implicit solution to an ODE is a relationship derived from the ODE in which you cannot solve for either variable, whereas an explicit solution is one in which you can solve for one of the variables.

I think what I would do is separate the variables to obtain:

\(\displaystyle y\,dy=-xe^x\,dx\)

Now integrate both sides:

\(\displaystyle \int_1^y u\,du=\int_x^0 ve^v\,dv\)

What do you find?
 
  • #3
Okay that's what I suspected. I got $c=1$ thus I got for my final explicit solution $y= \sqrt{2e^x-2xe^x-1}$
 
  • #4
shamieh said:
Okay that's what I suspected. I got $c=1$ thus I got for my final explicit solution $y= \sqrt{2e^x-2xe^x-1}$

I get the same. (Yes)
 

1. What is an explicit solution to an IVP?

An explicit solution to an initial value problem (IVP) is a function or equation that directly gives the value of the dependent variable at any point in the domain, without the need for further calculations or approximations. It is the most accurate and precise way of solving an IVP.

2. How do you find the explicit solution to an IVP?

To find the explicit solution to an IVP, you first need to use the initial conditions given in the problem to find the particular solution. Then, you can use integration techniques such as separation of variables or substitution to solve for the general solution. Finally, you can use the initial conditions again to determine the specific values of any constants and obtain the explicit solution.

3. What are the advantages of using an explicit solution to an IVP?

An explicit solution to an IVP is advantageous because it provides an exact, closed-form solution that is not dependent on any approximations or numerical methods. This allows for more accurate and precise predictions and calculations. It also provides a deeper understanding of the problem and its behavior.

4. Can there be multiple explicit solutions to an IVP?

Yes, there can be multiple explicit solutions to an IVP. This can occur when the initial conditions are not specific enough to determine a unique solution or when the problem has multiple solutions that satisfy the given conditions. In these cases, it is important to carefully consider the problem and choose the most appropriate solution.

5. Are there cases where an explicit solution to an IVP cannot be found?

Yes, there are some cases where it may not be possible to find an explicit solution to an IVP. This can happen when the problem is too complex or does not have a known solution in terms of elementary functions. In these cases, numerical methods or other approximation techniques may be used to obtain an approximate solution.

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