In problems 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form .
The given equation is the form of linear coefficients.
If the right-hand side of the equation can be expressed as a function of the ratio alone, then we say the equation is homogeneous.
Equations of the form
When the right-hand side of the equation can be expressed as a function of the combination , where a and b are constants, that is, then the substitution transforms the equation into a separable one.
A first-order equation that can be written in the form , where P(x) and Q(x) are continuous on an interval (a,b) and n is a real number, is called a Bernoulli equation.
We have used various substitutions for y to transform the original equation into a new equation that we could solve. In some cases, we must transform both x and y into new variables, say u and v. This is the situation for equations with linear coefficients-that is, equations of the form
The given equation satisfies
Therefore, the given equation is the form of a linear coefficient.
Question: Riccati Equation. An equation of the form (18) is called a generalized Riccati equation.
use the result of part (a) to find all the other solutions to this equation. (The particular solution can be found by inspection or by using a Taylor series method; see Section 8.1.)
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