The graph below shows solution curves of a differential equation. Find an equation for the curve through the given point.
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$$\frac { d y } { d x } = 1 - \frac { 6 } { 5 } x ^ { 1 / 5 }$$
A) $y = x + x ^ { 6 / 5 } + \frac { 3 } { 5 }$
B) $y = 1 - x ^ { 6 / 5 } - \frac { 3 } { 5 }$
C) $y = x - x ^ { 6 / 5 } + \frac { 5 } { 3 }$
D) $y = x - x ^ { 6 / 5 } + \frac { 3 } { 5 }$

Question

The graph below shows solution curves of a differential equation. Find an equation for the curve through the given point.
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$$\frac { d y } { d x } = 1 - \frac { 6 } { 5 } x ^ { 1 / 5 }$$
A) $y = x + x ^ { 6 / 5 } + \frac { 3 } { 5 }$ 
B) $y = 1 - x ^ { 6 / 5 } - \frac { 3 } { 5 }$ 
C) $y = x - x ^ { 6 / 5 } + \frac { 5 } { 3 }$ 
D) $y = x - x ^ { 6 / 5 } + \frac { 3 } { 5 }$

Quizplus · Accepted Answer

Comparing the given differential equation with the standard form $\frac{dy}{dx} = f(x)$, we can see that $f(x) = 1-\frac{6}{5}x^{\frac{1}{5}}$. Therefore, integrating both sides with respect to $x$, we get
$$y = \int\left(1 - \frac { 6 } { 5 } x ^ { 1 / 5 }\right)dx = x - 5\cdot\frac{5}{6}\cdot\frac{x^{\frac{6}{5}}}{\frac{6}{5}}+C,$$
where $C$ is the constant of integration. Simplifying the above expression, we get
$$y = x - x ^ { 6 / 5 } + C.$$
Using the fact that the curve passes through the point $(1,-1)$, we get
$$-1 = 1 - 1 ^ { 6 / 5 } + C,$$
which gives us $C = -\frac{3}{5}$. Therefore, the equation of the curve passing through the point $(1,-1)$ is
$$y = x - x ^ { 6 / 5 } - \frac { 3 } { 5 },$$
which is choice (D).

The Graph Below Shows Solution Curves of a Differential Equation dydx=1−65x1/5\frac { d y } { d x } = 1 - \frac { 6 } { 5 } x ^ { 1 / 5 }dxdy​=1−56​x1/5

The Graph Below Shows Solution Curves of a Differential Equation $\frac { d y } { d x } = 1 - \frac { 6 } { 5 } x ^ { 1 / 5 }$