In problems identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form .
The given equation is the form of both homogeneous and Bernoulli.
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 ax + by, where a and b are constants, that is, then the substitution z = ax + by 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.
Equation of Linear coefficients
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 .
Let u = y - 4x. Then, differentiate it to find the value of .
Integrating Equation (1),
Substitute u = y - 4x.
It seems that the given equation is linear coefficient.
Therefore, the given equation is the form of linear coefficient.
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