Separable differential equations Knowledge-Based Download PDF

Questions

1. 1.

   What is the particular solution to the differential equation $\frac{dy}{dx} = \frac{2xy}{x^2 + 1}$ satisfying the initial condition $y(1) = 6$?

   • A) $y = x^2 + 5$
   • B) $y = 3x^2 + 3$
   • C) $y = 6e^{x^2 - 1}$
   • D) $y = \frac{12}{x^2 + 1}$
2. 2.

   What is the particular solution to the differential equation $\frac{dy}{dx} = \frac{2x + 3}{y^2}$ satisfying the initial condition $y(0) = 2$?

   • A) $y = \sqrt[3]{3x^2 + 9x + 8}$
   • B) $y = \sqrt[3]{3x^2 + 9x + 2}$
   • C) $y = \sqrt{x^2 + 3x + 4}$
   • D) $y = \sqrt[3]{x^2 + 3x + 8}$
3. 3.

   What is the particular solution to the differential equation $\frac{dy}{dx} = \frac{\cos x}{e^y}$ satisfying the initial condition $y(0) = 0$?

   • A) $y = \ln(\cos x)$
   • B) $y = \sin x$
   • C) $y = \ln(\sin x)$
   • D) $y = \ln(\sin x + 1)$
4. 4.

   What is the particular solution to the differential equation $\frac{dy}{dx} = \frac{2x + 1}{2y}$ satisfying the initial condition $y(1) = 2$ for $y > 0$?

   • A) $y = \sqrt{x^2 + x + 4}$
   • B) $y = \sqrt{x^2 + x + 2}$
   • C) $y = \sqrt{2x^2 + x + 1}$
   • D) $y = x^2 + x + 2$
5. 5.

   What is the particular solution to the differential equation $\frac{dy}{dx} = 2x(y^2 + 1)$ satisfying the initial condition $y(0) = 1$?

   • A) $y = \tan(x^2 + 1)$
   • B) $y = \tan\left(2x^2 + \frac{\pi}{4}\right)$
   • C) $y = \sqrt{2e^{2x^2} - 1}$
   • D) $y = \tan\left(x^2 + \frac{\pi}{4}\right)$