Confusion about integration constant

[malignant]

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?

 

[Chestermiller]

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.

 

[malignant]

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.

 

[Chestermiller]

Solution 1: (y^3)/(3x^3) + ln(x) = 8/3

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

No difference!