Q. 20 - Calculus

Question

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.  

     $\frac{dy}{dx}=\frac{x+1}{\sqrt{x}}$

Step-by-Step Solution

Answer

Ans:   The solution to the differential equation $\frac{dy}{dx}=\frac{x+1}{\sqrt{x}}$ is $\frac{2}{3}{x}^{3/2}+2\sqrt{x}+C$

Step 1. Given information.

given,

       $\frac{dy}{dx}=\frac{x+1}{\sqrt{x}}$

Step 2. Consider the differential equation defined by equation (1) given below and solve it by using antidifferentiation and/or separation of the variable method.

 $\frac{dy}{dx}=\frac{x+1}{\sqrt{x}} \dots \dots \left(1\right)$

Step 3. Now,

Note that the differential equation $\left(1\right)$ does not contain the dependent variable at all, so technically the variables have already been separated. So, the differential equation can be solved by antidifferentiation. Thus, the solution of the differential equation is obtained by integrating both the sides

      $\begin{array}{r}\int dy=\int \frac{x+1}{\sqrt{x}}dx\\ =\int \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)dx\\ =\int \sqrt{x}dx+\int x^{-1/2}dx\\ =\frac{2}{3}{x}^{3/2}+2\sqrt{x}+C\end{array}$

Hence, a solution to the differential equation $\frac{dy}{dx}=\frac{x+1}{\sqrt{x}}$ is  $\frac{2}{3}{x}^{3/2}+2\sqrt{x}+C$.