Problem 7. A Bernoulli equation is a particular kind of first order ODE that can be written in the form x'(t) = a(t)x+ g(t)x" where some integer n #1, and a(t) are some suitably integrable functions. Using the substitution y(x) = x¹-n, we can actually rewrite this equation as an ODE involving the dependent variable y instead. Use this technique to solve the initial value problem x' (t) = 2 2t -x+ 3t X x(1) = 2 Hint: when rewriting the equation using your substitution, you should find a way to use the substitution so that the variable x does not appear at all-it should only contain y and t as variables. Once you find a solution, you should reexpress it in terms ofx and t. "
Problem 7. A Bernoulli equation is a particular kind of first order ODE that can be written in the form x'(t) = a(t)x+ g(t)x" where some integer n #1, and a(t) are some suitably integrable functions. Using the substitution y(x) = x¹-n, we can actually rewrite this equation as an ODE involving the dependent variable y instead. Use this technique to solve the initial value problem x' (t) = 2 2t -x+ 3t X x(1) = 2 Hint: when rewriting the equation using your substitution, you should find a way to use the substitution so that the variable x does not appear at all-it should only contain y and t as variables. Once you find a solution, you should reexpress it in terms ofx and t. "
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 7. A Bernoulli equation is a particular kind of first order ODE that can be written in the form x'(t) = a(t)x+ g(t)x"
where some integer n 1, and a(t) are some suitably integrable functions. Using the substitution y(x) = x¹−¹,
we can actually rewrite this equation as an ODE involving the dependent variable y instead. Use this technique
to solve the initial value problem
2 2t
= -x+
3t x
x(1) = 2
Hint: when rewriting the equation using your substitution, you should find a way to use the substitution so that
the variable x does not appear at all-it should only contain y and t as variables. Once you find a solution, you
should reexpress it in terms ofx and t.
x' (t)
"
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