Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-Differential equation: $x y ^ { \prime } + y = \frac { 2 x } { 5 }$
Initial condition: $\mathrm { y } ( 5 ) = 2$
Solution candidate: $y = \frac { 5 } { x } + \frac { x } { 5 }$

Question

Quizplus · Accepted Answer

To check if the given function is a solution of the differential equation, we need to substitute it into the differential equation and see if it satisfies the equation for all values of x.

Substituting y = $\frac{5}{x} + \frac{x}{5}$, we get:

$x \frac{dy}{dx} + \left(\frac{5}{x} + \frac{x}{5}\right) = \frac{2x}{5}$

Differentiating y with respect to x, we get:

$\frac{dy}{dx} = -\frac{5}{x^2} + \frac{1}{5}$

Substituting this into the above equation, we get:

$x \left(-\frac{5}{x^2} + \frac{1}{5}\right) + \left(\frac{5}{x} + \frac{x}{5}\right) = \frac{2x}{5}$

Simplifying this expression, we get:

$-\frac{5}{x} + 1 + \frac{x^2}{5x} = \frac{2x}{5x}$

$\frac{-25 + 5x^2 + x^3}{5x^2} = \frac{2}{5}$

Multiplying both sides by $5x^2$, we get:

$x^3 - 25x + 50 = 0$

This is not true for all values of x, and hence the given function is not a solution of the differential equation. Therefore, the answer is B.

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