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In one of my course textbook's examples, we are given this DE: $\frac{dy}{dx}=x + xy$, and also that $y(0)=1$. We are asked to find its particular solution.

So, in my textbook the solution was found by the 'method of solving a linear differential equation of the first-order'.

But I decided to solve it by the 'method of separation of variables'.

So after a couple of steps, I arrived at: $$\ln|1+y| = \frac{x^2}{2} + \ln|k|$$ $$\implies \ln\left|\frac{1+y}{k}\right| = \frac{x^2}{2}$$

Now, putting the values of $x$ & $y$ we get; $k=2$. So, $$ \ln\left|\frac{1+y}{2}\right| = \frac{x^2}{2}$$ $\implies$ $y = -1$ $\pm$ $2 (e)^{\dfrac{x^2}{2}} $

But according to my textbook the solution is (only) : $y = -1 + 2(e)^\dfrac{x^2}{2}$.

So did I do something wrong (And if so, what?)

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    $\begingroup$ Note that you have to not only find solutions to the differential equation, but also satisfy a given initial condition $\endgroup$ – AnotherJohnDoe Oct 12 '17 at 12:52
  • $\begingroup$ Yes, don't forget that $y(0) = 1$ $\endgroup$ – John Lou Oct 12 '17 at 12:53
  • $\begingroup$ @JohnLou, @AnotherJohnDoe; yes I did do that- that's how I got the value of k as 2. $\endgroup$ – Mr Reality Oct 12 '17 at 12:58
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$y=-1-2e^{\frac{x^2}{2}}$ is not a solution because $y(0)=-1-2e^0=-1-2*1=-3 \neq 1$

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  • $\begingroup$ So, if we hadn't been given that y(0)=1 then would $y= -1 - 2e^{\frac{x^2}{2}}$ be a solution too? $\endgroup$ – Mr Reality Oct 12 '17 at 13:45
  • $\begingroup$ I have some questions (you do not need to respond to each one if you don't want to) :- 1) For such questions, do we have to do what you did or/and check whether all the conditions are followed in order to get rid of the solutions which are not correct?; 2) I mean my textbook didn't mention why this solution is not correct, so did it just skip the step you''ve written ( and so is it something I'm expected to do?) or was there something in this particular solution (or the 'other' solution) because of which it was obvious? Thanks! $\endgroup$ – Mr Reality Oct 12 '17 at 13:55
  • $\begingroup$ I just wanted to show, in the simplest way possible, why the solution was incorrect. That doesn't mean this is the best way to answer a question (in a test or homework assignment). The best way to answer is what @Kevin did. I can't see what your textbook says, so I cannot comment on that. $\endgroup$ – Donat Pants Oct 12 '17 at 14:03
  • $\begingroup$ Then, can you answer my first comment- if it was not given that y(0)=1, then both of the obtained solutions would be correct, right? $\endgroup$ – Mr Reality Oct 12 '17 at 14:14
  • $\begingroup$ Not just that, the equation $$\ln|1+y| = \frac{x^2}{2} + \ln|k| => $$ $$y = -1 + k(e)^\dfrac{x^2}{2}$$ would be correct, you can put $$ k = 2 $$ or $$ k = -2 $$ and get your 2 solutions as private cases. $\endgroup$ – Donat Pants Oct 12 '17 at 14:16
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You should have, as your general solution, $$ \ln|1+y|=\frac{x^2}{2}+C\ \quad\iff\quad |1+y|=e^C e^{\frac{x^2}{2}} . $$

If $1+y>0$, you have the solution $$1+y= e^Ce^{\frac{x^2}{2}}\ \quad\iff\quad y=-1+ e^Ce^{\frac{x^2}{2}} . $$

If $1+y<0$, you have the solution $$-y-1= e^Ce^{\frac{x^2}{2}}\ \quad\iff\quad -y=1+ e^Ce^{\frac{x^2}{2}} . $$

In either case, the solution can be written as $y=-1+\widehat{C}e^{\frac{x^2}{2}} $, for some constant $\widehat{C}$ (different from the $C$ above).

Placing in the initial condition then specifies the particular solution.

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  • $\begingroup$ Ok, this seems amazing. So am I (..more generally, is one..) expected to solve these type of problems (e.g. finding particular solutions) like you've done? Because I see that if I had done what you did, then I wouldn't have needed to even use y(0)=1 to try to find the right solution. $\endgroup$ – Mr Reality Oct 12 '17 at 14:09
  • $\begingroup$ @MrReality Yes, well there are some folks who use the absolute value as a nicety to let the reader know that the $\ln$ cannot take negative values. With the absolute value though, there will always be two cases to consider. $\endgroup$ – Kevin Oct 12 '17 at 15:01

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