integration: if dy/dx = y/x, then what does y equal?

[soccerball3211]

If dy/dx=y/x then what does y equal?

I got y=-log(abs(x))+c

Can someone check my answer? Thank you for the help.

 

[stapel]

To check an integration answer, differentiate and see if you get what you'd started with.

. . . . .y = -ln(x) + c

. . . . .dy/dx = -(1/x)

. . . . .y/x = (-ln(x) + c)/x

Since this last does not equal -1/x, then there is a problem.

Please reply showing all of your steps. Thank you.

Eliz.

 

[soroban]

Hello, soccerball3211!

If \(\displaystyle \frac{dy}{dx}\:=\:\frac{y}{x}\), then what does \(\displaystyle y\) equal?

As Eliz. Stapel pointed out, you can check your answer yourself . . .

Separate the variables: \(\displaystyle \L\,\frac{dy}{y}\;=\;\frac{dx}{x}\)

Integrate: \(\displaystyle \L\,\int\frac{dy}{y}\;=\;\int\frac{dx}{x}\)

and we get: \(\displaystyle \L\,\ln(y) \;= \;\ln x\,+\,c \;= \;\ln(x)\,+\,\ln C\;=\;\ln(Cx)\)

Take anti-logs: \(\displaystyle \L\,y\;=\;Cx\)