Separable differential equations Knowledge-Based Practice

Questions

1. 1.

   What is the general solution to the differential equation $\frac{dy}{dx} = \frac{y \ln(y)}{x}$ for $x > 0$ and $y > 1$, where $C$ is an arbitrary positive constant?

   • A) $y = C \ln(x)$
   • B) $y = x^C$
   • C) $y = \ln(Cx)$
   • D) $y = e^{Cx}$
2. 2.

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

   • A) $y = \sqrt[3]{3\sin(x) + 27}$
   • B) $y = \sqrt[3]{3\sin(x) + 9}$
   • C) $y = \sqrt[3]{\sin(x) + 27}$
   • D) $y = \sqrt[3]{-3\cos(x) + 30}$
3. 3.

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

   • A) $y = \ln(x^3) + 1$
   • B) $y = \ln(x^3 + 1)$
   • C) $y = e^{x^3} - 1$
   • D) $y = \ln(x^3 - 1)$
4. 4.

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

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

   Find 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) $y = \tan\left(x^2 - \frac{\pi}{4}\right)$
   • D) $y = \tan\left(x^2 + \frac{\pi}{4}\right)$

Answers and step-by-step solutions appear after you finish the test.