Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-Differential equation: $x y ^ { \prime } + y = \frac { 2 x } { 9 }$
Initial condition: $y ( 9 ) = 2$
Solution candidate: $y = \frac { 9 } { x } + \frac { x } { 9 }$

Question

Quizplus · Accepted Answer

First, we need to differentiate the solution candidate to find $y'$. 
$$y = \frac { 9 } { x } + \frac { x } { 9 }$$
$$y' = -\frac{9}{x^2} + \frac{1}{9}$$
Now we substitute $y$ and $y'$ into the differential equation to check if it holds:
$$x y' + y = \frac{2x}{9}$$
$$x \left(-\frac{9}{x^2} + \frac{1}{9}\right) + \left(\frac { 9 } { x } + \frac { x } { 9 }\right) = \frac{2x}{9}$$
Simplifying this equation, we get:
$$-1 + \frac{1}{x^2} + 1 = \frac{2}{9}$$
$$\frac{2}{9} = \frac{2}{9}$$
Since this equation is true, we can conclude that the given function is indeed a solution of the differential equation. Furthermore, the initial condition $y(9) = 2$ is satisfied by the solution candidate, since 
$$y(9) = \frac{9}{9} + \frac{9}{9} = 2.$$ 
Therefore, we can confirm that the answer is A.

Determine If the Given Function Y = F(x) Is a Solution