Problem 53 - Calculus Volume 3

Question

Prove that if \(a, b,\) and \(c\) are positive constants, then all solutions to the second-order linear differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=0\) approach zero as \(x \rightarrow \infty .\) (Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)

Step-by-Step Solution

Answer

All solutions approach zero as \(x \to \infty\).

Step 1: Characteristic Equation

For the given second-order linear differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=0\), assume a solution of the form \(y=e^{rx}\). Substituting \(y\), \(y'\), and \(y''\) into the differential equation gives the characteristic equation \(ar^2 + br + c = 0\).

Step 2: Determine the Nature of the Roots

Solve the quadratic characteristic equation \(ar^2 + br + c = 0\) using the discriminant \(\Delta = b^2 - 4ac\). The nature of the roots is determined by the value of \(\Delta\):- If \(\Delta > 0\), there are two distinct real roots.- If \(\Delta = 0\), there is a repeated real root.- If \(\Delta < 0\), there are complex conjugate roots.

Step 3: Case 1 - Two Distinct Real Roots

If the roots are distinct and real (\(\Delta > 0\)), the general solution is \(y = C_1 e^{r_1 x} + C_2 e^{r_2 x}\), where \(r_1\) and \(r_2\) are negative because their coefficients of \(b\) and \(c\) are positive. As \(x \rightarrow \infty\), both exponential terms go to zero.

Step 4: Case 2 - Repeated Real Roots

If there is a repeated real root (\(\Delta = 0\)), the root is \(r = -\frac{b}{2a}\), which is negative. The general solution for repeated roots is \(y = (C_1 + C_2 x) e^{rx}\). Since \(r < 0\), the exponential term \(e^{rx}\) tends to zero as \(x \rightarrow \infty\).

Step 5: Case 3 - Complex Conjugate Roots

If the roots are complex conjugates (\(\Delta < 0\)), they are of the form \(r = \alpha \pm i\beta\), where \(\alpha = -\frac{b}{2a} < 0\). The solution is \(y = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x))\). The exponential factor \(e^{\alpha x}\) tends to zero as \(x \rightarrow \infty\) because \(\alpha < 0\).

Step 6: Conclusion

In all cases – distinct real roots, repeated real roots, and complex conjugate roots – the solutions to \(a y^{\prime \prime}+b y^{\prime}+c y=0\) tend to zero as \(x \rightarrow \infty\). This is due to the negative exponential factors that dominate the behavior of the solutions.

Key Concepts

Characteristic Equation

A key step in solving a second-order linear differential equation of the form \(a y'' + b y' + c y = 0\) is to assume a solution of the form \(y = e^{rx}\). This assumption helps us find the characteristic equation, which is a quadratic equation given by \(ar^2 + br + c = 0\). This characteristic equation provides the roots, which are critical in determining the behavior of solutions. By substituting \(y = e^{rx}\), \(y' = re^{rx}\), and \(y'' = r^2e^{rx}\) into the original differential equation, we derive the characteristic equation as \(ar^2 + br + c = 0\). Solving this quadratic equation yields the roots \(r_1\) and \(r_2\), which dictate the nature of the system's solutions.

Distinct Real Roots

When the characteristic equation has a positive discriminant (\(\Delta > 0\)), it indicates two distinct real roots. In this scenario, we find the general solution of the differential equation in the form \(y = C_1 e^{r_1 x} + C_2 e^{r_2 x}\), where \(r_1\) and \(r_2\) are the roots.

• Both roots will be real and distinct.
• In our exercise, because the original coefficients \(a, b,\) and \(c\) are positive, both \(r_1\) and \(r_2\) will be negative.
• This leads to the exponential terms \(e^{r_1 x}\) and \(e^{r_2 x}\) tending towards zero as \(x\) approaches infinity.

Thus, for distinct real roots, all components of the solution vanish to zero as \(x\) becomes very large, confirming exponential stability.

Complex Conjugate Roots

When the discriminant of our characteristic equation is negative (\(\Delta < 0\)), it results in complex conjugate roots. Complex conjugate roots are expressed in the form \(r = \alpha \pm i\beta\), where \(\alpha\) represents the real part and \(\beta\) the imaginary part.

• The general solution is expressed as \(y = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x))\).
• The exponential term \(e^{\alpha x}\) involves only the real part \(\alpha\).
• Given that \(\alpha\) is negative (as derived from \(-\frac{b}{2a}\)), \(e^{\alpha x}\) tends toward zero as \(x\) heads to infinity.

The oscillatory behavior due to \(\cos\) and \(\sin\) terms modulates the solution, but the overarching effect of the exponential term ensures that the solution still approaches zero, maintaining exponential stability.

Exponential Stability

Exponential stability in the context of differential equations refers to solutions that converge to zero as the independent variable approaches infinity. In our second-order linear differential equation with positive coefficients, exponential stability is assured across all types of roots due to their inherent negative real parts.

• Distinct real roots lead to solutions where each term independently decays to zero because of the negative exponents.
• Repeated real roots share similar properties, with solutions of the form \((C_1 + C_2 x)e^{rx}\) decaying to zero.
• Complex roots contribute an oscillatory component due to sine and cosine terms, but the negative real part of the exponential ensures overall decay.

In every case, the presence of negative real parts in the roots ensures that the solution's growth is restrained, and decay is favored. This negation of growth reflects the principle of exponential stability, which is crucial in system control and stability analysis.