Mathematics, 14.07.2019 01:10 lauramount
Use a substitution method to solve both of the following des, stating the general solution clearly, and showing all work clearly. a. dy/dx - y = e^xy^2 (solve explicitly for y.) b. dy/dx = x + y/x - y (you can leave the general solution in implicit form.)
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![\rightarrow \frac{dy}{dx}=x+\frac{y}{x}-y\\\\\rightarrow \frac{dy}{dx}-x=\frac{y}{x}-y\\\\\rightarrow \frac{dy}{dx}+y(1-\frac{1}{x})=x\\\\\text{Integrating factor}=e^{\int{1-\frac{1}{x}}\,dx}\\\\=e^{x-\log x}\\\\ \text{Multiplying both sides by} e^{x-\log x}\\\\e^{x-\log x}\times[\frac{dy}{dx}+y(1-\frac{1}{x})]=x \times e^{x-\log x}\\\\y\times e^{x-\log x} =\int x \times e^{x-\log x} \, dx}\\\\y\times e^{x-\log x}=\int x \times \frac{e^x}{e^{\log x}}\,dx\\\\y\times e^{x-\log x}=\int x \times \frac{e^x}{x} \, dx\\\\y\times e^{x-\log x}=e^x+K](/tpl/images/0086/8066/3c9ed.png)
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Use a substitution method to solve both of the following des, stating the general solution clearly,...
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