The left side is correct, but you made a mistake when you integrated xcos(x). You can check what you have by differentiating x2sin x + cos x. If your integration is correct, you should get d/dx(x2sin x + cos x) = xcosx.hangainlover said:Homework Statement
(3y^2 + 2y)y' = xcosx
xyy'=ln(x) ; y(1)=2
Homework Equations
The Attempt at a Solution
for the first one i get y^3 + y^2= (x^2) sin(x) +cos(x) + C (constant)
You might have made a typo here. The solution should be y2 = (ln x)2 + 4. It's probably OK to leave it this way, but if you need to, you can solve for y, as y(x) = +/-sqrt( (ln x)2 + 4)hangainlover said:and the second one is y^2 + ln(x)^2 +4
hangainlover said:the problem is there is no way i can define those two as function y of x or function x of y
because of the exponents.
So my question is, if a question is asking for a solution to a differential equation, can we just leave it as that above?
A differential equation is a mathematical equation that describes how a variable changes over time. It involves derivatives, which represent the rate of change of the variable. To solve a differential equation, we use mathematical techniques such as separation of variables, substitution, and integration.
The general rule for solving a differential equation is to isolate the variable with its derivatives on one side of the equation and all other terms on the other side. Then, we use mathematical techniques to find the function that satisfies the equation.
There are several types of differential equations, including ordinary differential equations, partial differential equations, and exact differential equations. Ordinary differential equations involve only single-variable functions, while partial differential equations involve multi-variable functions. Exact differential equations have a specific relationship between the variables and their derivatives.
Some common techniques for solving differential equations include separation of variables, substitution, integrating factors, and using the method of undetermined coefficients. Each technique has its own advantages and is used for different types of differential equations.
Differential equations are essential in science because they help us model and understand real-world phenomena, such as population growth, chemical reactions, and motion of objects. They also allow us to make predictions and solve problems in various fields, including physics, chemistry, biology, and engineering.