Using separation of variables, it is found that the expression is:
- [tex]f(x) = \pm \sqrt{x^3 - x + 16}[/tex]
The differential equation is:
[tex]\frac{dy}{dx} = \frac{3x^2 - 1}{2y}[/tex]
Solution by separation of variables
- Every term with y is placed on one side, every term with x on the other side, and then both sides are integrated.
Hence:
[tex]\frac{dy}{dx} = \frac{3x^2 - 1}{2y}[/tex]
[tex]2y dy = (3x^2 - 1) dx[/tex]
[tex]\int 2y dy = \int (3x^2 - 1) dx[/tex]
[tex]y^2 = x^3 - x + K[/tex]
[tex]y = \pm \sqrt{x^3 - x + K}[/tex]
Since f(1) = 4:
[tex]4 = \sqrt{1 - 1 + K}[/tex]
[tex]\sqrt{K} = 4[/tex]
[tex](\sqrt{K})^2 = 4^2[/tex]
[tex]K = 16[/tex]
Hence, the expression is:
[tex]f(x) = \pm \sqrt{x^3 - x + 16}[/tex]
To learn more about separation of variables, you can take a look at https://brainly.com/question/14318343
