Find the general solutions of the following equations: 𝐚 (d^2 y)/( d x^2)+5 (d y)/( d x)+6 y=0 𝐛

step 1 Here we are given the second order differential equation y prime prime plus 2y prime plus 5y is

Find the general solutions of the following equations: 𝐚...

Find the general solutions of the following equations: $\mathbf{a} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+5 \frac{\mathrm{d} y}{\mathrm{~d} x}+6 y=0$ $\mathbf{b} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+4 \frac{\mathrm{d} y}{\mathrm{~d} x}+4 y=0$ $\mathbf{c} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}+4 y=0$

Find the general solutions of the following equations:
$\mathbf{a} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+5 \frac{\mathrm{d} y}{\mathrm{~d} x}+6 y=0$
$\mathbf{b} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+4 \frac{\mathrm{d} y}{\mathrm{~d} x}+4 y=0$
$\mathbf{c} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}+4 y=0$

Key Concepts

Discriminant Analysis

The discriminant of the characteristic equation (obtained from the quadratic formula) indicates the nature of the roots: a positive discriminant implies two distinct real roots, zero implies a repeated real root, and a negative discriminant implies a pair of complex conjugate roots. Each case leads to a different structure for the general solution.

General Solution Forms

Depending on the roots of the characteristic equation, the general solution of the differential equation has different forms. For distinct real roots, the solution is a linear combination of two exponentials. For repeated real roots, the solution includes an additional polynomial factor multiplying the exponential term. For complex conjugate roots, the solution is represented in terms of real functions using sine and cosine expressions multiplied by an exponential decay term.

Second Order Linear Differential Equations

These are differential equations involving the second derivative of the unknown function and are linear in the dependent variable and its derivatives. Their coefficient functions are often constants in many standard problems, and they form the basis for many physical models including mechanical vibrations and electrical circuits.

Characteristic Equation

This is formed by assuming a solution of exponential form and substituting it into the differential equation. The substitution converts the differential equation into an algebraic equation (usually quadratic for second order differential equations) whose roots determine the behavior of the general solution.

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