Find the solution to the differential equation $x y ^ { \prime } - ( 6 + x ) y = 0$ with $y ( 1 ) = 1$ , $y ( x ) \geq 0$ for all x.
A) $y = x ^ { 7 } e ^ { x }$
B) $y = x ^ { 7 } e ^ { x - 1 }$
C) $y = x ^ { 6 } e ^ { x - 1 }$
D) $y = x ^ { 6 } e ^ { x } + 1 - e$

Question

Quizplus · Accepted Answer

This differential equation can be solved using the method of separating variables. Rearrange the equation to $\frac{dy}{y} = \frac{(6+x)}{x}dx$. Integrating both sides gives $\ln|y| = 6\ln|x| + x + C$. Exponentiating both sides yields $y = e^{6\ln|x| + x + C} = e^{6\ln|x|}e^{x}e^{C} = C'x^6e^x$, where $C'$ is a new constant. Using the initial condition $y(1) = 1$, we find $C' = e^{-1}$. Thus, the solution is $y = x^6e^{x-1}$.

Find the Solution to the Differential Equation xy′−(6+x)y=0x y ^ { \prime } - ( 6 + x ) y = 0xy′−(6+x)y=0

Find the Solution to the Differential Equation $x y ^ { \prime } - ( 6 + x ) y = 0$