Help with simplifying for a convenient general solution

[kiwilazer]

Hello I am studying differential equations and this problem is very basic. I am struggling to simply the natural logs using the exponential to ln rule[e^ln( )]. I want to solve the equation for y(x) . Thanks in advance.

 

[Deleted member 4993]

Hello I am studying differential equations and this problem is very basic. I am struggling to simply the natural logs using the exponential to ln rule[e^ln( )]. I want to solve the equation for y(x) . Thanks in advance.

Please share some of your struggles here - so that we know where to start.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.

 

[tkhunny]

[math]e^{ln(q)} = q[/math]
[math]10^{log_{10}(q)} = q[/math]
Where are you struggling with this?

 

[HallsofIvy]

You have, on one line, ln|y|= 4 ln|1+ x| and, on the next line, \(\displaystyle e^{4 ln|1+ x|}+ e^C\). That is incorrect. \(\displaystyle e^{a+ b}= (e^a)(e^b)\) NOT \(\displaystyle e^a+ e^b\), Also \(\displaystyle 4ln|1+ x|= ln|1+ x|^4\).

So this is \(\displaystyle e^{ln|y|}= \left(e^{ln|1+ x|^4}\right)e^{ln(C)}\)
\(\displaystyle e^{ln|y|}= |y|\), \(\displaystyle e^{ln|1+x|^4}= |1+ x|^4\), and \(\displaystyle e^{ln(C)}\) is just e to a constant so is a constant itself. We can call that C' (some would just use "C" again although it is a different constant).

So this is \(\displaystyle |y|= C'|1+ x|^4\).