$\text { Use integration to find the particular solution of the differential equation } \frac { d y } { d x } = \frac { \ln x } { x }$ which passes through the point $( 1 , - 3 )$.
A) $y = \ln x + 3$
B)$y = \frac { ( \ln x ) ^ { 2 } } { 2 } - 3$
C)$y = \frac { ( \ln x ) ^ { 2 } } { 2 } + 3$
D) $y = - 3 ( \ln x + 1 )$
E) $y = \ln x - 3$\

Question

Quizplus · Accepted Answer

Integrating $\frac{d y}{d x} = \frac{\ln x}{x}$ gives $y = \frac{(\ln x)^2}{2} + C$. Using the point $(1, -3)$, we find $C = -3$, so the solution is $y = \frac{(\ln x)^2}{2} - 3$.