Differential Equations Ordinary and Partial Differential Equations Math Forum

November 9th, 2016, 03:12 AM   #11
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Quote:
 Originally Posted by Mrkukas I get 1/2ln(2x+1)=ln(t/(t+1))+ln(C), but what should I do then?
$$\frac12 \ln{|2x+1|}=\ln{\left| \frac{t}{t+1} \right|}+\ln{C_1} \quad (C_1 \gt 0)$$

Note the absolute values that you should have from your integral. Combine the terms on the right and multiply by 2.

$$\ln{|2x+1|}=2\ln{\left| \frac{C_1 t}{t+1} \right|} \quad (C_1 \gt 0)$$

Use another logarithmic identity and eliminate the logarithms.

$$|2x+1|= \left| \frac{C_1 t}{t+1} \right|^2 \quad (C_1 \gt 0)$$

Bringing the exponent inside the absolute value bars, the right-hand side is now guaranteed to be positive, so they are redundant.

$$|2x+1|= \left( \frac{C_1 t}{t+1} \right)^2 = \frac{C_1^2 t^2}{(t+1)^2} \quad (C_1 \gt 0)$$

Now we remove the absolute value signs from the left-hand side.

$$2x+1 = \pm \frac{C_1^2 t^2}{(t+1)^2} \quad (C_1 \gt 0)$$

We can absorb the $\pm$ into the constant, setting $C=\pm C_1^2 \ne 0$.

$$2x+1 = \frac{C t^2}{(t+1)^2} \quad (C \ne 0)$$

Finally, since you divided by $(2x+1)$ when separating your variables, this work assumes that $2x+1\ne 0$. So now we put $x=-\frac12$, a constant function with a derivative of zero, into the original equation. Finding that the equation is satisfied, we note that if $2x+1=0$, the right-hand side of our working must also be zero, which we can achieve by allowing $C=0$. Thus

$$2x+1 = \frac{C t^2}{(t+1)^2} \quad (\text{for all C})$$

Last edited by skipjack; November 20th, 2016 at 07:18 AM. Tags differential, equation, order Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post millers4 Differential Equations 2 April 23rd, 2016 08:08 AM BonaviaFx Differential Equations 8 March 29th, 2015 08:20 AM crevoise Differential Equations 5 July 6th, 2012 12:51 AM jakeward123 Differential Equations 19 March 26th, 2011 12:50 PM Seng Peter Thao Differential Equations 0 June 30th, 2007 10:55 AM

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