A Bernoulli differential equation is one of the form dy/dx+P(x)=Q(x)y^n Observe

Question

A Bernoulli differential equation is one of the form

dy/dx+P(x)=Q(x)y^n

Observe that, if n=0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=y^{1-n} transforms the Bernoulli equation into the linear equation
du/dx+(1-n)P(x)u=(1-n)Q(x)

Use an appropriate substitution to solve the equation

xy'+y=-6xy^2

and find the solution that satisfies y(1)=5.

y(x)=?

Explanation / Answer

You switched the differential operators. The correct formulas are: dy/dx + P(x)·y = Q(x)·yn and du/dx + (1-n)·P(x)·u = (1-n)·Q(x) x·y' + y = - 8·x·y² y' + (1/x)·y = - 8·y² i.e. n = 2 P(x) = 1/x Q(x) = 8 So substitute u = y?¹ = 1/y and get: u' - (1/x)·y = 8 Such a 1st order inseparable DE of the type: y' - p(x)·y = q(x) can be solved using e raised to the power of antiderivative of p as integrating factor: µ = e^( ? p(x) dx) The solution is given by: y = ( ? µ·q(x) dx + C )/µ For the transformed DE p(x) = -1/x q(x) = 8 Hence: µ = e^( ? -1/x dx) = e^(-ln(x)) = 1/x => u = ( ? (1/x)·8 dx + C ) / (1/x) = ( 8·ln(x) + C )·x => y = 1/u = 1/[ ( 8·ln(x) + C )·x ] Apply boundary condition to evaluate C: y(1) = 9 1/[ ( 8·ln(1) + C )·1 ] = 9 1/C = 9 C = 1/9 => y = 1/[ ( 8·ln(x) + 1/9 )·x ] = 9/[ (72·ln(x) + 1 )·x ]

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