Confusion about integration constant

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SUMMARY

The discussion centers around the integration constant in the context of solving a differential equation (DE) represented as (y^3)/(3x^3) + ln(x) = c with the initial condition y(1) = 2. Two methods yield different values for the constant c: c = 8/3 when substituting directly and c = 8 when manipulating the equation by multiplying through by 3x^3. The discrepancy arises from the handling of the constant during the integration process, specifically the need to account for coefficients when combining constants with c.

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malignant
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After solving a DE I ended up with (y3)/(3x3) + ln(x) = c and initial conditions y(1) = 2.

If I just plug in straight away I get (23) / (3(13)) + ln(1) = c which is c = 8/3 but if I multiply the equation by 3x3 first: (y^3) + (3x3)ln(x) = (3x3)c and the 3 goes into the c and solving for c I get 23 + 3(1)3ln(1) = (1)3c and c = 8 which is the right answer.

I can see why these answers are different but I don't think I understand the constant. If I don't combine other constants with c I get an incorrect answer? When can I/do I have to combine constants with c?
 
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malignant said:
After solving a DE I ended up with (y3)/(3x3) + ln(x) = c and initial conditions y(1) = 2.

If I just plug in straight away I get (23) / (3(13)) + ln(1) = c which is c = 8/3 but if I multiply the equation by 3x3 first: (y^3) + (3x3)ln(x) = (3x3)c and the 3 goes into the c and solving for c I get 23 + 3(1)3ln(1) = (1)3c and c = 8 which is the right answer.

I can see why these answers are different but I don't think I understand the constant. If I don't combine other constants with c I get an incorrect answer? When can I/do I have to combine constants with c?
When you did it by the second method, you were missing a factor of 3 as the coefficient of c.
 
Chestermiller said:
When you did it by the second method, you were missing a factor of 3 as the coefficient of c.

Oh, that was because I combined it with the c.
 
Solution 1: (y^3)/(3x^3) + ln(x) = 8/3

Solution 2: (y^3) + (3x^3)ln(x) = 8(x^3)

No difference!
 

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