Question

Find the implicit solution of y′=(x2+3x+2)/(y−2),y(2)=1

Answer 1

Answer on Question #69633 – Math – Differential Equations

Question

$$y' = (x^2 + 3x + 2) / (y - 2), \quad y(2) = 1$$

Solution

$$\frac{dy}{dx} = \frac{x^2 + 3x + 2}{y - 2}$$ $$(y - 2)dy = (x^2 + 3x + 2)dx$$ $$\int (y - 2)dy = \int (x^2 + 3x + 2)dx$$ $$\frac{y^2}{2} - 2y = \frac{x^3}{3} + \frac{3}{2}x^2 + 2x + C$$

If $y(2) = 1$, then

$$\frac{1}{2} - 2 = \frac{2^3}{3} + \frac{3}{2}x^2 + 2x - C$$ $$C = -\frac{85}{6}$$

And the implicit solution of the differential equation is

$$\frac{y^2}{2} - 2y = \frac{x^3}{3} + \frac{3}{2}x^2 + 2x - \frac{85}{6}$$

Answer: $\frac{y^2}{2} - 2y = \frac{x^3}{3} + \frac{3}{2}x^2 + 2x - \frac{85}{6}$.

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