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I'm getting confused as to what to do with the C's.
\frac{{dy}}{{dx}} + 2xy = 0
Back of Book: y\left( x \right) = C\exp \left( { - x^2 } \right)
\begin{array}{l}<br /> \frac{{dy}}{{dx}} = - 2xy\,\,\,\, \Rightarrow \,\,\,\,\frac{{dy}}{y} = - 2x\,dx\,\,\,\, \Rightarrow \,\,\,\,\frac{1}{y}dy = - 2x\,dx \\ <br /> \\ <br /> \int_{}^{} {\frac{1}{y}dy} = \int_{}^{} { - 2x\,dx} \\ <br /> \\ <br /> \ln \left( y \right) = \frac{{ - 2x^2 }}{2} + C\,\,\,\, \Rightarrow \,\,\,\,\ln \left( y \right) = - x^2 + C\,\,\,\, \Rightarrow \,\,\,\, \\ <br /> \\ <br /> \exp \left( {\ln \left( {y + C} \right)} \right) = \exp \left( { - x^2 + C} \right)\,\,\,\, \Rightarrow \,\,\,\, \\ <br /> \\ <br /> y + C = \exp \left( { - x^2 + C} \right)\,\,\,\, \Rightarrow \,\,\,\,y = \exp \left( { - x^2 + C} \right) - C \\ <br /> \end{array}
The C from the y part is not necessarily the same C as from the x part, is it? How did they end up with only 1 C, and how did they get their C out of the parenthesis to end up multiplying by the right side?
Thanks in advance!
Homework Statement
\frac{{dy}}{{dx}} + 2xy = 0
Homework Equations
Back of Book: y\left( x \right) = C\exp \left( { - x^2 } \right)
The Attempt at a Solution
\begin{array}{l}<br /> \frac{{dy}}{{dx}} = - 2xy\,\,\,\, \Rightarrow \,\,\,\,\frac{{dy}}{y} = - 2x\,dx\,\,\,\, \Rightarrow \,\,\,\,\frac{1}{y}dy = - 2x\,dx \\ <br /> \\ <br /> \int_{}^{} {\frac{1}{y}dy} = \int_{}^{} { - 2x\,dx} \\ <br /> \\ <br /> \ln \left( y \right) = \frac{{ - 2x^2 }}{2} + C\,\,\,\, \Rightarrow \,\,\,\,\ln \left( y \right) = - x^2 + C\,\,\,\, \Rightarrow \,\,\,\, \\ <br /> \\ <br /> \exp \left( {\ln \left( {y + C} \right)} \right) = \exp \left( { - x^2 + C} \right)\,\,\,\, \Rightarrow \,\,\,\, \\ <br /> \\ <br /> y + C = \exp \left( { - x^2 + C} \right)\,\,\,\, \Rightarrow \,\,\,\,y = \exp \left( { - x^2 + C} \right) - C \\ <br /> \end{array}
The C from the y part is not necessarily the same C as from the x part, is it? How did they end up with only 1 C, and how did they get their C out of the parenthesis to end up multiplying by the right side?
Thanks in advance!