Question

Differential equations:
y'=ycosx/sin^2x

ylnx-xy'=o

7yy'-5e^x=0

Answer 1

1. separable
2. linear
3. separable

Answer 2

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
\[e^{lny}=e^{((lnx)^2)^\frac{1}{2}+c}\rightarrow y=x+c\]
\[3) 7y \frac{dy}{dx}-5e^x=0\rightarrow 7ydy=5e^xdx \rightarrow \int\limits_{}7ydy=\int\limits_{}5e^xdx \rightarrow\]
\[\frac{7}{2}y^2=5e^x+c \rightarrow y^2=\frac{10}{7}e^x+c \rightarrow y=\sqrt{\frac{10}{7}e^x+c}\]

Answer 3

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

Thank you.

Answer 4

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