Consider the Bernoulli differential equation y' + 3y = y^3. In order to convert this into a linear differential equation, we would make the substitution
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A Bernoulli differential equation is one of the form dxdy+P(x)y=Q(x)yn 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 dxdu+(1−n)P(x)u=(1−n)Q(x) Use an appropriate substitution to solve the equation y−x9y=y5x11 and find the solution that satisfies y(1)=1 y(x)=
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The Bernoulli differential equation is one of the form dy/dx + P(x)y = 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 y' - 6/x y = y^3/x^2, and find the solution that satisfies y(1) = 1. y(x) =
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A Bernoulli differential equation is one of the form dy/dx + P(x)y = 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 = -9xy^2, and find the solution that satisfies y(1) = 8. y(x) =
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