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(1)y = q(1)y" and is called Bernoulli's equation after Jakob Bernoulli. If n ± 0, 1, then the substitution v = y'-" reduces Bernoulli's equation to a linear equation. Solve the given Bernoulli equation by using this substitution. Py + 4ty – y = 0, t > 0 y = ± + cr* 5t y = ± + c18 y = ± :+ c+ 2 y = ± 9t y = +V+ c 9t - ㅇ ㅇ
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(1)y = q(1)y" and is called Bernoulli's equation after Jakob Bernoulli. If n ± 0, 1, then the substitution v = y'-" reduces Bernoulli's equation to a linear equation. Solve the given Bernoulli equation by using this substitution. Py + 4ty – y = 0, t > 0 y = ± + cr* 5t y = ± + c18 y = ± :+ c+ 2 y = ± 9t y = +V+ c 9t - ㅇ ㅇ
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Bernoulli's Equation
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 \( n \neq 0, 1 \), then the substitution \( v = y^{1-n} \) reduces Bernoulli's equation to a linear equation.
Solve the given Bernoulli equation by using this substitution.
\[ t^2 y' + 4ty - y^3 = 0, \, t > 0 \]
### Multiple Choice Solutions:
1. \( y = \pm \sqrt{\frac{1}{5t} + ct^4} \)
2. \( y = \pm \frac{1}{\sqrt{2+ct^8}} \)
3. \( y = \pm \sqrt{\frac{1}{5t} + ct^4} \)
4. \( y = \pm \frac{2}{9t + ct^8} \)
5. \( y = \pm \sqrt{\frac{2}{9t + ct^8}} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cac1a6-5ad9-4f72-bdd9-21a202e53df4%2Ffac5b42a-73e6-4afa-8972-de549d5edd42%2F4n1sspd8_processed.png&w=3840&q=75)
Transcribed Image Text:### Bernoulli's Equation
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 \( n \neq 0, 1 \), then the substitution \( v = y^{1-n} \) reduces Bernoulli's equation to a linear equation.
Solve the given Bernoulli equation by using this substitution.
\[ t^2 y' + 4ty - y^3 = 0, \, t > 0 \]
### Multiple Choice Solutions:
1. \( y = \pm \sqrt{\frac{1}{5t} + ct^4} \)
2. \( y = \pm \frac{1}{\sqrt{2+ct^8}} \)
3. \( y = \pm \sqrt{\frac{1}{5t} + ct^4} \)
4. \( y = \pm \frac{2}{9t + ct^8} \)
5. \( y = \pm \sqrt{\frac{2}{9t + ct^8}} \)
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