Exercise
$\frac{dy}{dx}=\frac{3x+y}{3x+y+1}$
Step-by-step Solution
Learn how to solve problems step by step online. Solve the differential equation dy/dx=(3x+y)/(3x+y+1). When we identify that a differential equation has an expression of the form Ax+By+C, we can apply a linear substitution in order to simplify it to a separable equation. We can identify that 3x+y has the form Ax+By+C. Let's define a new variable u and set it equal to the expression. Isolate the dependent variable y. Differentiate both sides of the equation with respect to the independent variable x. Now, substitute 3x+y and \frac{dy}{dx} on the original differential equation. We will see that it results in a separable equation that we can easily solve.
Solve the differential equation dy/dx=(3x+y)/(3x+y+1)
Final answer to the exercise
$\frac{1}{4}\left(3x+y\right)+\frac{3}{16}-\frac{3}{16}\ln\left(4\left(3x+y\right)+3\right)+\frac{1}{4}\ln\left(4\left(3x+y\right)+3\right)=x+C_0$