sorry I made a mistake in the original equation and now its fixed.andrewkirk said:The First term in your 2nd line is wrong, and the error propagates through from there.
what do you mean by left hand side ?andrewkirk said:You've expressed the right-hand side in terms of v, so that's progress. Now you need to do similar work on the left hand side.
Using ##v=\frac{y}{x}##, express ##\frac{dv}{dx}## in terms of y, y' and x, then see if you can use that to express y' in terms of v, v' and x. Equate that to the RHS and then with any luck you'll be able to use separation of variables to get an expression to be integrated over x on one side and the same for v on the other.
hey man, thanks for your time and help, I was able to get it resolved and it matched the answer in my book.andrewkirk said:LHS of the diff equation in section 1 of your OP, which is: ##\frac{dy}{dx}##.
A homogeneous differential equation is a type of differential equation where all the terms are of the same degree. This means that the dependent variable and its derivatives are the only variables present in the equation. The general form of a homogeneous differential equation is dy/dx = f(x,y).
The main difference between a homogeneous and non-homogeneous differential equation is the presence of a constant term. In a homogeneous equation, there are no constant terms, while in a non-homogeneous equation, there is at least one constant term present. This constant term affects the solution of the equation and makes it more complicated to solve.
To solve a homogeneous differential equation, you can use the method of separation of variables, substitution, or integrating factors. The specific method used will depend on the form of the equation and the techniques you are familiar with. It is important to note that not all homogeneous differential equations can be solved analytically, and in some cases, numerical methods may be required.
Homogeneous differential equations are widely used in various fields of science, including physics, chemistry, and engineering. They are used to model and describe many natural phenomena, such as population growth, chemical reactions, and motion of objects. Solving these equations helps us understand and predict the behavior of systems in the natural world.
Yes, there are many real-life applications of homogeneous differential equations. They are used in various fields of science and engineering to model and solve problems, such as in circuit analysis, heat transfer, and fluid mechanics. They are also used in economics and biology to study population dynamics and growth. Homogeneous differential equations are an essential tool in understanding and analyzing complex systems in the real world.