20 Jun 2025

Homogeneous Differential Equations with Constant Coefficients

Characteristic roots and the complete real solutions for distinct, repeated, and complex roots.

bsc semester-ii mathematical-physics constant-coefficient-equations characteristic-roots

Consider the homogeneous constant-coefficient equation

\[a_n\frac{d^ny}{dx^n}+a_{n-1}\frac{d^{n-1}y}{dx^{n-1}} +\cdots+a_1\frac{dy}{dx}+a_0y=0, \qquad a_n\ne0.\]

The exponential $y=e^{mx}$ is useful because every differentiation only multiplies it by $m$:

\[\frac{d^k}{dx^k}e^{mx}=m^ke^{mx}.\]

Substitution gives

\[e^{mx}\left(a_nm^n+a_{n-1}m^{n-1}+\cdots+a_1m+a_0\right)=0.\]

Since $e^{mx}\ne0$, the allowed values of $m$ satisfy the characteristic equation

\[\boxed{a_nm^n+a_{n-1}m^{n-1}+\cdots+a_1m+a_0=0}.\]

An $n$th-order equation requires $n$ linearly independent basis solutions and therefore carries $n$ arbitrary constants. Root multiplicity determines which factors of $x$ are needed to complete that basis.

Distinct real roots

For

\[y^{\prime\prime}-5y^{\prime}+6y=0,\]

the characteristic polynomial is

\[m^2-5m+6=(m-2)(m-3).\]

The two independent solutions are $e^{2x}$ and $e^{3x}$, so

\[\boxed{y=C_1e^{2x}+C_2e^{3x}}.\]

Repeated roots

If a root $m_0$ occurs $s$ times, the $s$ independent solutions are

\[e^{m_0x},\ xe^{m_0x},\ldots,x^{s-1}e^{m_0x}.\]

For example,

\[y^{\prime\prime}-4y^{\prime}+4y=0\]

has $(m-2)^2=0$, and therefore

\[\boxed{y=(C_1+C_2x)e^{2x}}.\]

To see why the factor $x$ appears, start with the distinct-root quotient

\[\frac{e^{(m_0+\varepsilon)x}-e^{m_0x}}{\varepsilon}.\]

Taking $\varepsilon\to0$ gives

\[\frac{\partial}{\partial m_0}e^{m_0x}=xe^{m_0x}.\]

Complex-conjugate roots

With real coefficients, a complex root $m=\alpha+i\beta$ is accompanied by $\alpha-i\beta$. Euler’s formula gives

\[e^{(\alpha+i\beta)x}=e^{\alpha x} \bigl(\cos\beta x+i\sin\beta x\bigr).\]

Taking real linear combinations of the conjugate solutions yields

\[\boxed{y=e^{\alpha x} \left(C_1\cos\beta x+C_2\sin\beta x\right)}.\]

For $y^{\prime\prime}+4y=0$, $m=\pm2i$, so

\[\boxed{y=C_1\cos2x+C_2\sin2x}.\]

Each result is verified by direct differentiation in the Unit II Maxima worksheet.

Solved Problems

1. A third-order initial-value problem

Solve

\[y^{\prime\prime\prime}-y^{\prime}=0, \qquad y(0)=1, \qquad y^{\prime}(0)=0, \qquad y^{\prime\prime}(0)=2.\]

The characteristic equation is

\[m^3-m=m(m-1)(m+1)=0.\]

The distinct roots $0,1,-1$ give

\[y=C_0+C_1e^x+C_2e^{-x}.\]

At $x=0$ the three conditions give

\[C_0+C_1+C_2=1, \qquad C_1-C_2=0, \qquad C_1+C_2=2.\]

Hence $C_1=C_2=1$ and $C_0=-1$, so

\[\boxed{y=2\cosh x-1}.\]

Direct differentiation gives $y^{\prime\prime\prime}-y^{\prime}=0$ and reproduces all three initial values. The three constants before applying the data agree with the order of the equation.

2. Complex roots with a decaying envelope

Solve

\[y^{\prime\prime}+2y^{\prime}+5y=0, \qquad y(0)=1, \qquad y^{\prime}(0)=0.\]

The characteristic polynomial is

\[m^2+2m+5=0,\]

so $m=-1\pm2i$. Therefore

\[y=e^{-x}(C_1\cos2x+C_2\sin2x).\]

The first condition gives $C_1=1$. Differentiating,

\[y^{\prime}(0)=-C_1+2C_2=0,\]

so $C_2=1/2$. Thus

\[\boxed{y=e^{-x}\left(\cos2x+\frac12\sin2x\right)}.\]

Substitution gives a zero differential-equation residual. The oscillation is bounded by an envelope proportional to $e^{-x}$, so the solution tends to zero as $x\to\infty$.

Descriptive Questions

1. Derive the characteristic equation for an $n$th-order homogeneous differential equation with constant coefficients.
2. Explain why a root of multiplicity $s$ generates the factors $1,x,\ldots,x^{s-1}$.
3. Show how a complex-conjugate root pair produces a real sine-cosine basis.
4. Relate the order of the equation, the number of independent solutions, and the number of initial conditions required for a unique solution.

Numerical Problems

1. For $y^{\prime\prime}-6y^{\prime}+13y=0$, determine the roots and the change in the exponential envelope between $x=0$ and $x=1$. Final answer: $m=3\pm2i$; the envelope is multiplied by $e^3=20.0855$.
2. Solve $y^{\prime\prime}+6y^{\prime}+9y=0$ with $y(0)=2$, $y^{\prime}(0)=-3$, and evaluate $y(1)$. Final answer: $y=(2+3x)e^{-3x}$; $y(1)=5e^{-3}=0.24894$.
3. A mode obeys $y^{\prime\prime}+25y=0$. Find its angular frequency and period if the independent variable is time in seconds. Final answer: $\omega=5\,\mathrm{rad\,s^{-1}}$ and $T=2\pi/5=1.25664\,\mathrm{s}$.
4. Construct the monic differential equation whose characteristic roots are $-2,-2,1$. Final answer: $(m+2)^2(m-1)=m^3+3m^2-4$; hence $y^{\prime\prime\prime}+3y^{\prime\prime}-4y=0$.

The residuals, initial values, and numerical evaluations are verified in the Unit II problem-check worksheet.

References

1. Linear differential equation — Wikipedia
2. William E. Boyce, Richard C. DiPrima, and Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 11th ed., chapter “Higher-Order Linear Equations,” Wiley.
3. Erwin Kreyszig, Advanced Engineering Mathematics, 10th ed., Chapter 2, “Second-Order Linear ODEs,” Wiley.
4. Mary L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Chapter 8, “Ordinary Differential Equations,” Wiley.