Consider the differential equation and the solution Find the particular solution satisfying the initial conditions

Question

Consider the differential equation  and the solution  Find the particular solution satisfying the initial conditions

Quizplus · Accepted Answer

The given differential equation is separable. We can write it as:

$$\frac{dy}{dx}=\frac{y-2\sqrt{xy}}{x}$$

Let $\sqrt{x}=u$, then $\frac{1}{2\sqrt{x}}=\frac{1}{2u}$, and $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=\frac{1}{2u}\cdot\frac{dy}{du}$.

Substituting these in the given differential equation, we get:

$$\frac{1}{2u}\cdot\frac{dy}{du}=\frac{y-2u\cdot\sqrt{y}}{u^2}$$

$$\Rightarrow 2y\frac{dy}{du}-4u\sqrt{y}=y\frac{du}{dx}$$

This equation can be solved by separating the variables and integrating both sides. Integrating the left side with respect to y and the right side with respect to x, we get:

$$\int 2y dy = \int\frac{y}{u} du-2\int u^{1/2}du$$

$$\Rightarrow y^2 = 2y\sqrt{x}-\frac{4}{3}x^{3/2}+C$$

where C is the constant of integration.

Using the initial condition $y(1)=2$, we get:

$$2^2 = 2\cdot 2\cdot 1^{1/2}-\frac{4}{3}\cdot 1^{3/2}+C$$

$$\Rightarrow C=2$$

Therefore, the particular solution satisfying the given initial condition is:

$$y^2 = 2y\sqrt{x}-\frac{4}{3}x^{3/2}+2$$

Squaring both sides and simplifying, we get:

$$y = \frac{2}{\sqrt{x}+1}$$

Hence, the correct choice is (B).

Consider the Differential Equation and the Solution Find