2:59 Previous Problem Problem List Next Problem A Bernoulli differential equation is one of the form dy +p(x)y = q(x)y" (*) dx Observe that, if n = 0 or n = 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 + (1 − n)p(x)u = (1 − n)q(x). Consider the initial value problem xy' + y = 6xy², y(1) = −2. This differential equation can be written in the form (*) with 1 p(x) = help (formulas) X q(x) = 6, and help (formulas) n = 2 help (numbers) The substitution u = y-1 will transform it into the linear equation du + u = -6 help (formulas) dx Using the substitution above, we rewrite the initial condition in terms of x and u: 1 u(1) : = help (numbers) 2 Now solve the linear equation for u, and find the solution that satisfies the initial condition. u(x) = -6xln(x) - 1x help (formulas) 2:59 Previous Problem Problem List Next Problem du dx + (1 − n)p(x)u = (1 − n)q(x). Consider the initial value problem xy' + y = 6xy², y(1) = −2. This differential equation can be written in the form (*) with 1 p(x) = 44 help (formulas) q(x) = = 6, and help (formulas) n = 2 help (numbers) The substitution u = y-1 will transform it into the linear equation du + u = -6 help (formulas) dx Χ Using the substitution above, we rewrite the initial condition in terms of x and u: 1 u(1) = help (numbers) 2 Now solve the linear equation for u, and find the solution that satisfies the initial condition. u(x) = -6x ln(x) - 1x help (formulas) Finally, solve for y. Y(r) = help (formulas) Book: Section 1.5 of Notes on Diffy Qs