Apply Theorems 5 and 6 to find general solutions of the differential equations. Primes denote derivatives with respect to x. 1. 4y''+4y'+y=0 2. 9y''-12y'+4y=0
Apply Theorems 5 and 6 to find general solutions of the differential equations. Primes denote derivatives with respect to x. 1. 4y''+4y'+y=0 2. 9y''-12y'+4y=0
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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Apply Theorems 5 and 6 to find general solutions of the differential equations. Primes denote derivatives with respect to x.
1. 4y''+4y'+y=0
2. 9y''-12y'+4y=0
![**THEOREM 6: Repeated Roots**
If the characteristic equation in (18) has equal (necessarily real) roots \( r_1 = r_2 \), then
\[ y(x) = (c_1 + c_2 x) e^{r_1 x} \quad \text{(21)} \]
This theorem describes the general solution form for differential equations when the characteristic equation has repeated roots. The expression includes constants \( c_1 \) and \( c_2 \), emphasizing the need for an additional term \( c_2 x \) due to the multiplicity of the root to accommodate the solution space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F160e110e-0605-48f5-87d6-7df540ee4abb%2Fc4520faa-2159-4509-9f53-2d9012d41364%2Fiuxudgin_processed.png&w=3840&q=75)
Transcribed Image Text:**THEOREM 6: Repeated Roots**
If the characteristic equation in (18) has equal (necessarily real) roots \( r_1 = r_2 \), then
\[ y(x) = (c_1 + c_2 x) e^{r_1 x} \quad \text{(21)} \]
This theorem describes the general solution form for differential equations when the characteristic equation has repeated roots. The expression includes constants \( c_1 \) and \( c_2 \), emphasizing the need for an additional term \( c_2 x \) due to the multiplicity of the root to accommodate the solution space.
![**Theorem 5: Distinct Real Roots**
If the roots \( r_1 \) and \( r_2 \) of the characteristic equation in (18) are real and distinct, then
\[
y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}
\]
(19)
This theorem provides the general solution to a linear differential equation with constant coefficients, given that the characteristic equation has real and distinct roots. Here, \( c_1 \) and \( c_2 \) are constants determined by initial conditions, and \( e \) is the base of the natural logarithm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F160e110e-0605-48f5-87d6-7df540ee4abb%2Fc4520faa-2159-4509-9f53-2d9012d41364%2F2o3x4do_processed.png&w=3840&q=75)
Transcribed Image Text:**Theorem 5: Distinct Real Roots**
If the roots \( r_1 \) and \( r_2 \) of the characteristic equation in (18) are real and distinct, then
\[
y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x}
\]
(19)
This theorem provides the general solution to a linear differential equation with constant coefficients, given that the characteristic equation has real and distinct roots. Here, \( c_1 \) and \( c_2 \) are constants determined by initial conditions, and \( e \) is the base of the natural logarithm.
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