5.2 Solutions to Differential Equations with SICY-O

A separable differential equation is a type of differential equation that can be expressed in the form $$\frac{dy}{dx} = g(x)h(y)$$ , allowing the variables to be separated on opposite sides of the equation.

To separate the equation, rewrite it as $$\frac{1}{y}dy=(2-x)dx$$ .

The general solution is obtained by integrating both sides after separation, typically resulting in an equation involving an arbitrary constant C.

The first step is to rearrange the equation to isolate $$dy$$ and $$dx$$ on opposite sides.

The integration yields $$\text{ln}|y|=2x-\frac{1}{2}x^2+C$$ .

To find a particular solution, substitute the given initial conditions into the general solution to solve for the constant C.

The constant C represents the family of solutions to the differential equation, accounting for different initial conditions.

Integrating gives $$\frac{3}{2}y^2 = x^2 + x + C$$ .

The method of integrating factors is used for non-separable differential equations, where a function is multiplied to both sides to make the left side a derivative.

Initial conditions are used to determine the specific solution from the general solution by providing values for the variables.