Help with the Lambert W function

[adriel2004]

Hello, I need help to solve a differential equation.
I know the solution is involving the Lambert W function but I‘m stuck. Here‘s the equation:

(1+x)ln(1+x)+3.2x=44.8

According to Wolfram Alpha, the solution should be :



If you could help me with the steps, I‘d be grateful, thank you all!
Regards

 

[blamocur]

Why do you call it differential equation? I don't see any derivatives in there.

 

[blamocur]

Substitution [imath]v = \frac{1}{1+x}[/imath] looks promising to me.

 

[Dr.Peterson]

Hello, I need help to solve a differential equation.
I know the solution is involving the Lambert W function but I‘m stuck. Here‘s the equation:

(1+x)ln(1+x)+3.2x=44.8

According to Wolfram Alpha, the solution should be :

View attachment 38349

If you could help me with the steps, I‘d be grateful, thank you all!
Regards

One way to figure out what they did would be to work backward: Rewrite the equation to the form W(...) = ..., and then think about how that relates to the given equation.

If you prefer to pretend you haven't seen this answer, then start with the definition of W: Since it is the inverse of f(w) = we^w, you'd like to obtain an expression of that sort. How about letting u = ln(1+x)?

 

[mario99]

Hello, I need help to solve a differential equation.
I know the solution is involving the Lambert W function but I‘m stuck. Here‘s the equation:

(1+x)ln(1+x)+3.2x=44.8

According to Wolfram Alpha, the solution should be :

View attachment 38349

If you could help me with the steps, I‘d be grateful, thank you all!
Regards

First look at the answer given by W|A and think. Why the answer has the number [imath]48[/imath]? This means that the first step is to add [imath]3.2[/imath] to both sides.

[imath] (1+x)\ln(1+x)+3.2x + 3.2 = 44.8 + 3.2[/imath]

How can this help you to go further?

 

[mario99]

Why do you call it differential equation? I don't see any derivatives in there.

He meant that he has solved a differential equation and arrived to this step where he could not isolate [imath]x[/imath]. It is a common step as we have seen this before in the @logistic_guy differential equation if you remember!