Question

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

$\frac{dy}{dx}=\frac{3x+1}{xy}$

Short answer

Ans: The solution of the differential equation $\frac{dy}{dx}=\frac{3x+1}{xy}$is $y=\pm \sqrt{6x+2\ln \left|x\right|+C}$

Step-by-step solution

Step 1. Given information.

given,

$\frac{dy}{dx}=\frac{3x+1}{xy}$

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.      

$\frac{dy}{dx}=\frac{3x+1}{xy}......\left(1\right)$

Step 3. Solution

Note that the differential equation (1) is of the form of $\frac{dy}{dx}=p\left(x\right)q\left(y\right)$in which $p\left(x\right)=\frac{3x+1}{x}$and $q\left(y\right)=\frac{1}{y}$. So the differential equation can be solved by applying the variable separable method. Separate the variables and integrate both the sides

$\begin{array}{r}\int ydy=\int \frac{3x+1}{x}dx\\ \frac{y^2}{2}=\int 3dx+\int \frac{1}{x}dx\\ =3x+\ln \left|x\right|+C_1\\ y^2=6x+2\ln \left|x\right|+2C_1\end{array}$

Replace the constant localid="1649177740460" $2C_1$by another constant $C$and write the solution as $y^2=6x+2\ln \left|x\right|+C$

Hence a solution to the differential equation $\frac{dy}{dx}=\frac{3x+1}{xy}$is localid="1649177691002" $y=\pm \sqrt{6x+2\ln \left|x\right|+C}$.