- #1
Jeann25
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How would I start this one? I tried dividing by y, but that does not work
y'=xy/(1+x²)
y'=xy/(1+x²)
Jeann25 said:How would I start this one? I tried dividing by y, but that does not work
y'=xy/(1+x²)
Jeann25 said:Here's what I did:
1/y y' = x/(1+x²)
ln y = 1/2 ln (1+x²)+c
y = ce^(1/2 ln(1+x²))
Answer's supposed to be: c*sqrt(1+x²)
A separable differential equation is a type of ordinary differential equation (ODE) in which the dependent variable and independent variable can be separated into two separate functions. This allows for the equation to be solved by integrating both sides separately.
The main technique used to solve separable differential equations is separation of variables. This involves isolating the dependent and independent variables on opposite sides of the equation, and then integrating both sides. Other techniques such as substitution and partial fractions may also be used in some cases.
The initial value in a separable differential equation is a specific set of conditions given at the starting point of the problem. This includes the initial value of the dependent variable and the value of the independent variable at the starting point. These values are used to find the particular solution to the equation.
Sure! For example, let's say we have the separable differential equation dy/dx = x/y with the initial condition y(1) = 2. First, we can rewrite the equation as ydy = xdx. Then, we can integrate both sides to get (y^2)/2 = (x^2)/2 + C. Plugging in the initial value, we get (2^2)/2 = (1^2)/2 + C, which simplifies to C = 1. Therefore, the particular solution to the equation is (y^2)/2 = (x^2)/2 + 1.
Separable differential equations are used to model a variety of natural phenomena in fields such as physics, chemistry, biology, and economics. For example, they can be used to describe the rate of growth of a population, the spread of an infectious disease, or the cooling of a hot object. They are also commonly used in engineering applications to predict the behavior of systems and optimize designs.