QuestionGiven the differential equation $\frac{d y}{d x}=\frac{3 x+1}{2 y}$, find the particular solution, $y=f(x)$, with the initial condition $f(2)=-4$.
Answer Attempt 1 out of 2 $y=$ $\square$ Submit Answer

Studdy Solution
Substitute $C = 8$ back into the equation:
$$y^2 = \frac{3}{2}x^2 + x + 8$$
Solve for $y$:
$$y = \pm \sqrt{\frac{3}{2}x^2 + x + 8}$$
Since $f(2) = -4$, we choose the negative root:
$$y = -\sqrt{\frac{3}{2}x^2 + x + 8}$$
The particular solution is:
$$y = -\sqrt{\frac{3}{2}x^2 + x + 8}$$