A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.


d. For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C₁cosh(kt) + C₂sinh(kt), where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively (Section 7.3).

Verified step by step guidance

1. Start with the given differential equation: \(y''(t) - k^{2} y(t) = 0\), where \(k > 0\).
2. Write the characteristic equation associated with the differential equation: \(r^{2} - k^{2} = 0\).
3. Solve the characteristic equation for \(r\): \(r^{2} = k^{2}\), so \(r = \pm k\).
4. Write the general solution using the exponential functions corresponding to the roots: \(y(t) = C_{1} e^{kt} + C_{2} e^{-kt}\).
5. Recall the definitions of hyperbolic cosine and sine: \(\cosh(kt) = \frac{e^{kt} + e^{-kt}}{2}\) and \(\sinh(kt) = \frac{e^{kt} - e^{-kt}}{2}\). Use these to rewrite the general solution as \(y(t) = A \cosh(kt) + B \sinh(kt)\), where \(A\) and \(B\) are constants related to \(C_{1}\) and \(C_{2}\).

Key Concepts

Second-Order Linear Differential Equations

These are differential equations involving the second derivative of a function and can often be solved by finding characteristic equations. Solutions typically involve exponential, trigonometric, or hyperbolic functions depending on the nature of the roots of the characteristic equation.

Characteristic Equation and Its Roots

For the equation y'' - k²y = 0, the characteristic equation is r² - k² = 0, yielding roots r = ±k. These roots determine the form of the general solution, which can be expressed using exponentials or hyperbolic functions when roots are real and distinct.

Hyperbolic Functions (cosh and sinh)

Hyperbolic cosine and sine, defined as cosh(t) = (e^t + e^{-t})/2 and sinh(t) = (e^t - e^{-t})/2, provide an alternative way to express solutions of differential equations with real roots. They are especially useful for rewriting solutions involving exponentials in a more compact and interpretable form.