Question

Find the general solution of each of the following differential equations. $\frac{dy}{dx}=\frac{3y}{3y^{2/3}-x}$

Short answer

\frac{3}{5} y^{5/3} - \frac{1}{2} y^2 = 3x + C

Step-by-step solution

Identify the form of the differential equation

The given differential equation is $\frac{dy}{dx}=\frac{3y}{3y^{2/3}-x}$. It can be identified as a separable differential equation.

Rewrite the equation to separate variables

To separate the variables, rewrite the equation: $\frac{dy}{dx}=\frac{3y}{3y^{2/3}-x}$. Multiply both sides by $3y^{2/3}-x$ and $dx$ to get $(3y^{2/3}-x)dy=3ydx$.

Simplify and further separate the variables

Rearrange terms to isolate all terms involving $y$ on one side and all terms involving $x$ on the other. $(3y^{2/3}-x)dy=3ydx$. Simplify to $(3y^{2/3}dy-ydy)=3dx$.

Integrate both sides

Integrate both sides to solve for $y$. For the left-hand side, the integral becomes $\frac{3}{5}{y}^{5/3}-\frac{1}{2}{y}^2$. For the right-hand side, the integral is $3x+C$. Combining these, we get $\frac{3}{5}{y}^{5/3}-\frac{1}{2}{y}^2=3x+C$.

Simplify the general solution

Combine and simplify the constants to write the general solution. The final form of the general solution will be $\frac{3}{5}{y}^{5/3}-\frac{1}{2}{y}^2=3x+C$.

Key concepts

differential equations

A differential equation is an equation that involves unknown functions and their derivatives. These equations play a crucial role in various fields like physics, engineering, and economics.
By finding solutions to these equations, we can predict how systems change over time. There are different types of differential equations, but one of the simplest forms is a first-order differential equation.
In this exercise, we are dealing with a first-order differential equation represented as $\frac{dy}{dx}=\frac{3y}{3y^{2/3}-x}$.
Our goal is to find the general solution, which helps us understand how the dependent variable, y, changes concerning the independent variable, x.

integration

Integration is a fundamental concept in calculus that can be thought of as the reverse process of differentiation. In the context of differential equations, integration is used to solve equations involving derivatives.
When we integrate, we effectively find the original function given its derivative. For example, if we have $dy=f(x)dx$, integrating both sides with respect to x will give us the original function y in terms of x.
In our exercise, after separating the variables, we perform integration on both sides of the equation. Specifically, we integrate $3y^{2/3}-y$ with respect to y and 3 with respect to x. The integration results help us combine the terms to form the general solution.

variables separation

Separation of variables is a method for solving first-order differential equations. The idea is to rearrange the equation so that all terms involving the dependent variable (y) are on one side, and all terms involving the independent variable (x) are on the other.
For our differential equation $\frac{dy}{dx}=\frac{3y}{3y^{2/3}-x}$, we multiply both sides by $3y^{2/3}-x$ and $dx$ to separate the variables. This step yields $(3y^{2/3}-x)dy=3ydx$.
Next, we isolate the terms to get $3y^{2/3}dy-ydy=3dx$. This format allows us to integrate both sides easily.

general solution

The general solution to a differential equation is a family of functions that includes an arbitrary constant, usually denoted as C. This constant represents an infinite number of possible solutions, each corresponding to a different initial condition.
After performing the necessary integrations, we get a combined solution involving the constant C. In our exercise, after integrating both sides, we obtain $\frac{3}{5}{y}^{5/3}-\frac{1}{2}{y}^2=3x+C$. This is the general solution to our differential equation.
This form provides a complete description of all possible solutions and how y changes concerning x for various initial conditions.