In this article, we will explore how to solve homogeneous linear second-order differential equations with constant coefficients. In the previous article, we discussed the distinction between homogeneous and non-homogeneous equations. As part of this ongoing series, we'll now delve deeper into the methods of solving homogeneous equations, starting with the characteristic equation approach, and providing step-by-step examples to solidify your understanding.

1. Standard Form of a constant-coefficient linear second-order homogeneous equation:

\frac{d^2y}{dx^2} + a\frac{dy}{dx} + by = 0,

where $a$ and $b$ are constants.

1.1. Characteristic equation.

Generally when solving we consider the function $y = e^{mx}$ , and then we take derivatives

\frac{dy}{dx} = me^{mx} \quad \text{and} \quad \frac{d^2y}{dx^2} = m^2e^{mx}.

Substituting these into the equation:

m^2e^{mx} + am e^{mx} + b e^{mx} = 0.

Factoring out $e^{mx}$ :

e^{mx}(m^2 + am + b) = 0.

Since $e^{mx} \neq 0$ , the characteristic equation is:

m^2 + am + b = 0.

This equation is called the Auxiliary equation or Characteristic equation.

1.2. Steps to obtain the characteristic equation.

The characteristic equation of a constant coefficient linear ODE can be easily obtained by simply replacing the y and its derivatives by powers of m.

Example 1:

Consider the differential equation:

\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 0.

In the ODE, replace $\frac{d^2y}{dx^2}$ by $m^2$ , $\frac{dy}{dx}$ by m, and y by 1. Then we thus get the characteristic equation:

m^2 - 3m + 2 = 0

Example 2:

Consider the differential equation:

\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0.

In the ODE, replace $\frac{d^2y}{dx^2}$ by $m^2$ , $\frac{dy}{dx}$ by m, and y by 1. Then we thus get the characteristic equation:

m^2 - 4m + 4 = 0

Example 3:

Consider the differential equation:

\frac{d^2y}{dx^2} + y = 0.

In the ODE, replace $\frac{d^2y}{dx^2}$ by $m^2$ and y by 1. Then we thus get the characteristic equation:

m^2 + 1 = 0

1.3. Solutions to constant-coefficient 2nd-order linear Homogeneous ODEs.

Suppose you have the following characteristic equation:

am^2 + bm + c = 0.

We break it into three cases as follows.

Case I: Characteristic equation has two distinct real roots $m_1$ and $m_2$

The general solution of the ODE is:

y = Ae^{m_1x} + Be^{m_2x},

where $A$ and $B$ are constants.

Case II: Equal real roots $m$

In the case of repeated roots, the general solution of the ODE is:

y = Ae^{mx}x + Be^{mx}

Case III: Complex conjugate roots $m = p \pm iq$

The general solution is given by:

where $p$ and $q$ are real numbers.

So as above these cases cover all possibilities for the roots of the characteristic equation, providing a systematic approach to solving homogeneous second-order linear ODEs.

2. Example Problems

2.1. Example 1

\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 0

Solution Steps:

1. Form the characteristic equation:

m^2 + 5m + 6 = 0

(Replacing the derivative terms and y by powers of m).

2. Solve the characteristic equation:

Factorize $m^2 + 5m + 6$ :

(m + 2)(m + 3) = 0

Roots are $m_1 = -2$ , $m_2 = -3$ (distinct real roots).

3. Write the general solution:

The general solution for distinct real roots is:

y(x) = C_1 e^{-2x} + C_2 e^{-3x}

where $C_1$ and $C_2$ are constants determined by initial conditions (if given).

2.2. Example 2

\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0

Solution Steps:

1. Form the characteristic equation:

(Replacing the derivative terms and y by powers of m)

m^2 + 4m + 4 = 0

2. Solve the characteristic equation:

Factorize $m^2 + 4m + 4$ :

(m + 2)^2 = 0

Roots are $m = -2$ (repeated real root).

3. Write the general solution:

For repeated real roots, the general solution is:

y(x) = C_1e^{-2x} x + C_2e^{-2x}

where $C_1$ and $C_2$ are constants.

2.3. Example 3

\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 5y = 0

Solution Steps:

1. Form the characteristic equation:

(Replacing the derivative terms and y by powers of m).

m^2 - 2m + 5 = 0

2. Solve the characteristic equation:

Use the quadratic formula:

m = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)} = \frac{2 \pm \sqrt{4 - 20}}{2}= \frac{2 \pm \sqrt{-16}}{2} = 1 \pm 2i

(Complex roots).

3. Write the general solution:

For complex roots $m = \alpha \pm \beta i$ , the general solution is:

y(x) = e^{\alpha x} (C_1 \cos \beta x + C_2 \sin \beta x)

Here $\alpha = 1$ and $\beta = 2$ , so:

y(x) =C_1 e^x \cos 2x + C_2e^x \sin 2x

2.4. Example 4

\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 2y = 0

Solution Steps:

1. Form the characteristic equation:

(Replacing the derivative terms and y by powers of m).

m^2 + 2m + 2 = 0

2. Solve the characteristic equation:

Use the quadratic formula:

m = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(1)}}{2(1)} = \frac{-2 \pm \sqrt{4 - 4}}{2} = \frac{-2 \pm \sqrt{-4}}{2} = -1 \pm i

(Complex roots).

3. Write the general solution:

For complex roots $m = \alpha \pm \beta i$ , the general solution is:

y(x) = e^{\alpha x} (C_1 \cos \beta x + C_2 \sin \beta x)

Here $\alpha = -1$ and $\beta = 1$ , so:

y(x) = C_1 e^{-x}\cos x + C_2e^{-x} \sin x

2.5. Example 5

\frac{d^2y}{dx^2} + 6\frac{dy}{dx} + 9y = 0

1. Characteristic equation: $m^2 + 6m + 9 = 0$

2. Factorize: $(m + 3)^2 = 0$

3. Root: $m = -3$ (repeated).

4. General solution: $y(x) = C_1 e^{-3x} x + C_2 e^{-3x}$

2.6. Example 6

\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 13y = 0

1. Characteristic equation: $m^2 - 4m + 13 = 0$

2. Solve using quadratic formula:

m = \frac{4 \pm \sqrt{16 - 52}}{2} = 2 \pm 3i

3. General solution: $y(x) = C_1 e^{2x}\cos 3x + C_2e^{2x} \sin 3x$

2.7. Example 7

\frac{d^2y}{dx^2} + 9y = 0

1. Characteristic equation: $m^2 + 9 = 0$

2. Roots: $m = \pm 3i$

3. General solution: $y(x) = C_1 \cos 3x + C_2 \sin 3x$

2.8. Example 8

\frac{d^2y}{dx^2} + 7\frac{dy}{dx} + 12y = 0

1. Characteristic equation: $m^2 + 7m + 12 = 0$

2. Factorize: $(m + 3)(m + 4) = 0$

3. Roots: $m_1 = -3$ , $m_2 = -4$

4. General solution: $y(x) = C_1 e^{-3x} + C_2 e^{-4x}$

2.9. Example 9

\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 10y = 0

1. Characteristic equation: $m^2 - 2m + 10 = 0$

2. Roots: $m = 1 \pm 3i$

3. General solution: $y(x) =C_1 e^x\cos 3x + C_2 \ e^x \sin 3x$

2.10. Example 10

\frac{d^2y}{dx^2} + 4y = 0

1. Characteristic equation: $m^2 + 4 = 0$

2. Roots: $m = \pm 2i$

3. General solution: $y(x) = C_1 \cos 2x + C_2 \sin 2x$

These examples cover different cases of second-order linear ODEs with constant coefficients, including distinct real roots, repeated real roots, and complex roots.

3. Practice Question Set

Find the general solution of the following equations

$\frac{d^2y}{dx^2} - 2\frac{dy}{dx} - 15y = 0$
$\frac{d^2y}{dx^2} + 6\frac{dy}{dx} + 9y = 0$
$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 5y = 0$
$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} = 0$

Not sure about your answers? Don't worry, click the link below to check the video where answers are explained.

Practice Question Set Discussion

In conclusion, we've explored the method of solving homogeneous linear second-order differential equations using the characteristic equation approach. By working through examples, we’ve seen how the nature of the characteristic roots determines the form of the solution. In the next article, we’ll shift our focus to non-homogeneous equations, where we will introduce techniques like the method of undetermined coefficients to tackle these more complex systems.

Remember, the key to success in ODEs is practice and persistence. The examples and practice problems provided here are just the beginning—keep solving more problems to strengthen your understanding.

Stay tuned as we continue to build on our understanding of differential equations.

Related Articles

14. Linear homogeneous second-order differential equations.

1. Standard Form of a constant-coefficient linear second-order homogeneous equation:

1.1. Characteristic equation.

1.2. Steps to obtain the characteristic equation.

1.3. Solutions to constant-coefficient 2nd-order linear Homogeneous ODEs.

2. Example Problems

2.1. Example 1

2.2. Example 2

2.3. Example 3

2.4. Example 4

2.5. Example 5

2.6. Example 6

2.7. Example 7

2.8. Example 8

2.9. Example 9

2.10. Example 10

3. Practice Question Set