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July 8th, 2017, 11:29 AM   #1
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Constant C in differential equation

Hi guys,

$$\int {}\frac{1}{ky}dy =\int dx \rightarrow \frac{1}{k}\ln \left | y \right |=x+C$$

Why is C only on the right side and not on the left side?

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July 8th, 2017, 11:33 AM   #2
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You can put a 'c' on both sides if you want. C is an arbitrary constant. You can combine all constants to form one final constant.

$$\int {}\frac{1}{ky}dy =\int dx \rightarrow \frac{1}{k}\ln \left | y \right | + C1 =x+C2$$

$$\rightarrow \frac{1}{k}\ln \left | y \right | =x+C2 - C1$$

$$\rightarrow C2 - C1 = C$$

$$\rightarrow \frac{1}{k}\ln \left | y \right | =x+C $$
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Last edited by skipjack; July 8th, 2017 at 08:45 PM.
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July 10th, 2017, 04:18 AM   #3
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Might also be worth keeping in mind that the general solution of an $nth$ order differential equation will contain $n$ arbitrary constants.
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Last edited by skipjack; July 10th, 2017 at 04:46 AM.
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