(a) For what values of r does the function y = ex satisfy the differential equation 7y" + 34y' - 5y = 0? (Enter your answers as a comma-separated list.) r= (b) If r₁ and r₂ are the values of r that you found in part (a), show that every member of the family of functions y = ae1* + be 2* is also a solution. Let r₁ be the larger value and r₂ be the smaller value. We need to show that every member of the family of functions y = ae1* + be 2* is a solution of the differential equation 7y" + 34y' - 5y = 0. We must decide whether 7f"(x) + 34f'(x) - 5f(x) = 0 for f(x) = ae1* + be 2*. Substitute your values for ₁ and ₂ into f(x), then find f'(x) and f"(x). We have f'(x) = and f"(x) = Substituting and combining like terms, we get the following. 7f"(x) + 34f'(x) - 5f(x) = e²₁x + ( [ 1x Derzx Therefore, f(x) = ae1* + be 2* is a solution to the differential equation 7y" + 34y' - 5y = 0.
(a) For what values of r does the function y = ex satisfy the differential equation 7y" + 34y' - 5y = 0? (Enter your answers as a comma-separated list.) r= (b) If r₁ and r₂ are the values of r that you found in part (a), show that every member of the family of functions y = ae1* + be 2* is also a solution. Let r₁ be the larger value and r₂ be the smaller value. We need to show that every member of the family of functions y = ae1* + be 2* is a solution of the differential equation 7y" + 34y' - 5y = 0. We must decide whether 7f"(x) + 34f'(x) - 5f(x) = 0 for f(x) = ae1* + be 2*. Substitute your values for ₁ and ₂ into f(x), then find f'(x) and f"(x). We have f'(x) = and f"(x) = Substituting and combining like terms, we get the following. 7f"(x) + 34f'(x) - 5f(x) = e²₁x + ( [ 1x Derzx Therefore, f(x) = ae1* + be 2* is a solution to the differential equation 7y" + 34y' - 5y = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(a) For what values of r does the function y = ex satisfy the differential equation 7y" + 34y' - 5y = 0? (Enter your answers as a comma-separated list.)
r =
(b) If r₁ and ₂ are the values of r that you found in part (a), show that every member of the family of functions y = ae¹¹× + be^2× is also a solution.
1
Let ₁ be the larger value and r₂ be the smaller value. We need to show that every member of the family of functions y =
equation 7y" + 34y' - 5y = 0.
We must decide whether 7f"(x) + 34f'(x) − 5f(x) = 0 for f(x) ae 1x + be 2x.
Substitute your values for r₁ and
r₂
into f(x), then find f'(x) and f"(x).
We have f'(x) =
and f"(x)
=
Substituting and combining like terms, we get the following.
7f"(x) + 34f'(x) - 5f(x)
₁²₁x + ( [
Therefore, f(x) = ae^1× + be^2× is a solution to the differential equation 7y" + 34y' - 5y = 0.
er 2x
ae1x + be 2* is a solution of the differential](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdcd72f34-3ec5-4815-9094-73aadfcc33e2%2F8abbcc3c-c7e9-4f6e-8af3-6e2cf10b9331%2Fialfhnw_processed.png&w=3840&q=75)
Transcribed Image Text:(a) For what values of r does the function y = ex satisfy the differential equation 7y" + 34y' - 5y = 0? (Enter your answers as a comma-separated list.)
r =
(b) If r₁ and ₂ are the values of r that you found in part (a), show that every member of the family of functions y = ae¹¹× + be^2× is also a solution.
1
Let ₁ be the larger value and r₂ be the smaller value. We need to show that every member of the family of functions y =
equation 7y" + 34y' - 5y = 0.
We must decide whether 7f"(x) + 34f'(x) − 5f(x) = 0 for f(x) ae 1x + be 2x.
Substitute your values for r₁ and
r₂
into f(x), then find f'(x) and f"(x).
We have f'(x) =
and f"(x)
=
Substituting and combining like terms, we get the following.
7f"(x) + 34f'(x) - 5f(x)
₁²₁x + ( [
Therefore, f(x) = ae^1× + be^2× is a solution to the differential equation 7y" + 34y' - 5y = 0.
er 2x
ae1x + be 2* is a solution of the differential
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