Why Muss with the “+C”?

Here is a way to find the particular solution of a separable differential equation without using the +C and it might even be a little faster. As an example consider the initial value problem

\displaystyle \frac{dy}{dx}=\frac{y-1}{{{x}^{2}}}\text{ with }x\ne 0\text{ and }y\left( 2 \right)=0

Separate the variables as usual and then write each side a definite integral with the lower limit of integration the number from the initial condition and the upper limit variable. I’ll replace the dummy variable in the integrand with an upper case X or Y to avoid having x and y in two places.

\displaystyle \int_{0}^{y}{\frac{1}{Y-1}dY}=\int_{2}^{x}{{{X}^{-2}}dX}


\displaystyle \left. \ln \left| Y-1 \right| \right|_{0}^{y}=\left. -{{X}^{-1}} \right|_{2}^{x}

\displaystyle \ln \left| y-1 \right|-\ln \left| 0-1 \right|=-\frac{1}{x}+\frac{1}{2}

Note that near the initial condition where y = 0, \left( y-1 \right)<0 so \left| y-1 \right|=-\left( y-1 \right)=1-y . Continue and solve for y.

\displaystyle {{e}^{\ln \left( 1-y \right)}}={{e}^{-\frac{1}{x}+\frac{1}{2}}}

\displaystyle 1-y={{e}^{-\frac{1}{x}+\frac{1}{2}}}

\displaystyle y=1-{{e}^{-\frac{1}{x}+\frac{1}{2}}},\quad x>0

No muss; no fuss.


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