If y=y(x) is the solution curve of the differential equation [(x^2-4) d y-(y^2-3 y) d x=0,…

Question

If $y=y(x)$ is the solution curve of the differential equation $$\left(x^2-4\right) \mathrm{d} y-\left(y^2-3 y\right) \mathrm{d} x=0, x>2, y(4)=\frac{3}{2}$$ and the slope of the curve is never zero, then the value of $y(10)$ equals :

Accepted Answer

$$\begin{aligned}
& \left(x^2-4\right) d y-\left(y^2-3 y\right) d x=0 \
& \Rightarrow \int \frac{d y}{y^2-3 y}=\int \frac{d x}{x^2-4} \
& \Rightarrow \frac{1}{3} \int \frac{y-(y-3)}{y(y-3)} d y=\int \frac{d x}{x^2-4} \
& \Rightarrow \frac{1}{3}(\ln |y-3|-\ln |y|)=\frac{1}{4} \ln \left|\frac{x-2}{x+2}\right|+C \
& \Rightarrow \frac{1}{3} \ln \left|\frac{y-3}{y}\right|=\frac{1}{4} \ln \left|\frac{x-2}{x+2}\right|+C
\end{aligned}$$

Solutions · 1