## anonymous 5 years ago Differential equations: y'=ycosx/sin^2x ylnx-xy'=o 7yy'-5e^x=0

1. anonymous

1. separable 2. linear 3. separable

2. anonymous

you could use separable for all three: $1) \frac{dy}{dx}=\frac{ycosx}{\sin^2x}\rightarrow \frac{dy}{y}=\frac{cosx}{\sin^2x}dx$ $\int\limits_{}\frac{dy}{y}=\int\limits_{}\frac{cosx}{\sin^2x}dx \rightarrow lny=-\frac{1}{sinx}+c$ use u substitution $e^{lny}=e^{-\frac{1}{sinx}+c}\rightarrow y=\frac{1}{e^{\frac{1}{sinx}}}+c$ $2) ylnx-x \frac{dy}{dx}=0\rightarrow \frac{dy}{y}=\frac{lnx}{x}dx$ $\int\limits_{}\frac{dy}{y}=\int\limits_{}\frac{lnx}{x}dx \rightarrow lny=\frac{1}{2}(lnx)^2+c \rightarrow$ use u substitution $e^{lny}=e^{((lnx)^2)^\frac{1}{2}+c}\rightarrow y=x+c$ $3) 7y \frac{dy}{dx}-5e^x=0\rightarrow 7ydy=5e^xdx \rightarrow \int\limits_{}7ydy=\int\limits_{}5e^xdx \rightarrow$ $\frac{7}{2}y^2=5e^x+c \rightarrow y^2=\frac{10}{7}e^x+c \rightarrow y=\sqrt{\frac{10}{7}e^x+c}$

3. anonymous

nadeem: An excellent presentation. Very professional. May I ask you what software you use to effect that result. Thank you.

4. anonymous

Thanks..... I used latex, the same software included in this website as an equation editor.