Sometimes it is possible to solve a nonlinear equation by making a change of the

Question

Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form y' + p(t)y = q(t)y^n and is called Bernoulli's equation after Jakob Bernoulli. If n notequalto 0, 1, then the substitution v = y^1-n reduces Bernoulli's equation to a linear equation. Solve the given Bernoulli equation by using this substitution. y' = 4/t y + t Squareroot y Use a capital C for the constant of integration and decimals rather than fractions. y(t) = _____

Explanation / Answer

n=1/2

v=y^{1-1/2}

v^2=y

y=v^2

y'=2vv'

Substituting gives

2vv'=4v^2/t+tv

2v'=4v/t+t

v'-2v/t=t/2

Integrating factor is

1/t^2

Multiplying gives

(v/t^2)'=1/(2t)

Integrating gives

v/t^2=ln(t)/2+C

v=t^2 ln(t)/2+C ln(t)

y^{1/2}=t^2 ln(t)/2+C ln(t)

y=(t^2 ln(t)/2+C ln(t))^2

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