Solve the Bernoulli equations. x 2 y + 2xy = y 3 HISTORICAL BIOGRAPHY James Bernoulli (1654-1705)

Solve the Bernoulli equations.

x²y′ + 2xy = y³

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HISTORICAL BIOGRAPHY James Bernoulli (1654-1705) A Bernoulli differential equation is of the form dy dx Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y¹-n transforms the Bernoulli equation into the linear equation du dx For example, in the equation + P(x)y= Q(x)y". or, equivalently, + (1 − n)P(x)u = (1 − n)Q(x). dy dx we have n = 2, so that u = y¹-2 = y¹ and du/dx = -y-² dy/dx. Then dy/dx = -y² du/dx = − u² du/dx. Substitution into the original equation gives u²¹ = ex u ² -u -2 du dx du dx · y = exy² +u = −ex. This last equation is linear in the (unknown) dependent variable u.