A Bernoulli differential equation is one of the form d y d x + P ( x ) y = Q ( x
ID: 3416732 • Letter: A
Question
A Bernoulli differential equation is one of the form dydx+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?ntransforms the Bernoulli equation into the linear equation dudx+(1?n)P(x)u=(1?n)Q(x).
Use an appropriate substitution to solve the equation xy?+y=?9xy2,
and find the solution that satisfies y(1)=?2.
y(x)= . A Bernoulli differential equation is one of the form dydx+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?ntransforms the Bernoulli equation into the linear equation dudx+(1?n)P(x)u=(1?n)Q(x).
Use an appropriate substitution to solve the equation xy?+y=?9xy2,
and find the solution that satisfies y(1)=?2.
y(x)= . dydx+P(x)y=Q(x)yn. dudx+(1?n)P(x)u=(1?n)Q(x). xy?+y=?9xy2,
Explanation / Answer
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A Bernoulli differential equation is on of the form
dy/dx + P(x)y = Q(x)yn
Observe that, if n = 0 or 1, the Bernoulli equation is linear. For the other values of n, the substitution y = y1-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 = -6xy2,
and find the solution that satisfies y(1) = -1.
Answer
xy' + y = -6xy^2
Divide each term by x
y' + (1/x)y = -6y^2
Now divide by y^2
y'/y^2 + 1/(xy) = -6
Let u = 1/y so u' = du/dx = du/dy dy/dx = -1/y^2*dy/dx = -y'/y^2
Substituting we have
-u' + (1/x)u = -6
u' - (1/x)u = 6
This is a linear differential equation.
Let v = 1/x so v' = -1/x^2 so v'/v = -1/x
Now multiply by the integrating factor v
u'v - (1/x)uv = 6v
u'v + (v'/v)uv = 6v
u'v + v'u = 6v
The LHS is just the product rule applied to uv - integrating we get
uv = 6ln(x) + C <--- remember v = 1/x
u = 6xln(x) + Cx
1/y = x[6ln(x) + C]
y = 1/{x[6ln(x) + C]}
When x = 1 y = -1
c=1
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