Solution method for [[Partial Differential Equations (PDEs)|PDEs]] Can be performed on ***separable*** equations: $ F(y,x)=M(y)\,N(x) $ $F(y,x)$ can be separated into a function of $x$ alone and a function of $y$ alone. Separate variables into terms: $ \frac{1}{M(y)}\frac{dy}{dx}=N(x) $ Then integrate: $ \int\frac{1}{M(y)}\frac{dy}{\cancel{dx}}\,\cancel{dx}=\int N(x)\,dx $ $ \int\frac{1}{M(y)}dy=\int N(x)\,dx $ # Example Separate variables into separate terms: $ \begin{align} \frac{dy}{dt}+2ty&=0\\ \frac{1}{y}\frac{dy}{dt}+2t&=0 \end{align} $ Then integrate: $ \begin{align} \int\frac{1}{y}\frac{dy}{dt}+2t\,dt&=\int0\,dt\\ \int\frac{1}{y}\frac{dy}{dt}\,dt+\int2t\,dt&=\int0\,dt\\ \int\frac{1}{y}\frac{dy}{\cancel{dt}}\,\cancel{dt}+t^2+c_1&=c_2 \end{align} $ $ \begin{gather} \ln|y|=\hat{c}-t^2\\ |y|=e^{\hat{c}-t^2}=e^\hat{c}e^{-t^2} \end{gather} $