Apply Theorems 5 and 6 to find general solutions of the differential equations. Primes denote derivatives with respect to x.1. 4y''+4y'+y=02. 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. **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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Answered: Apply Theorems 5 and 6 to find general…

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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Question

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.

**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.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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