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Solve the following.
Solve the differential equation: x(x - 1)(dy)/(dx) - y = x²(x - 1)². Find the particular solution…
Solve the differential equation: $x(x - 1)\dfrac{dy}{dx} - y = x^2(x - 1)^2$. Find the particular solution satisfying the condition $y = 0$ when $x = 0$.
Answer
Answer
Divide by $x(x - 1)$: $\dfrac{dy}{dx} - \dfrac{1}{x(x - 1)}\,y = x(x - 1)$.
Integrating factor $= e^{-\int \frac{dx}{x(x - 1)}} = e^{-\ln\left|\frac{x - 1}{x}\right|} = \dfrac{x}{x - 1}$.
$y\cdot\dfrac{x}{x - 1} = \int x(x - 1)\cdot\dfrac{x}{x - 1}\,dx = \dfrac{x^3}{3} + C$, so $y = \dfrac{x - 1}{x}\left(\dfrac{x^3}{3} + C\right)$.
When $x = 0$, $y = 0$ requires $C = 0$ (otherwise $y$ is not defined at $x = 0$).
Particular solution: $y = \dfrac{x^2(x - 1)}{3}$.
Final answer: $y = \dfrac{x^2(x - 1)}{3}$