Higher order linear differential equations notes — Unit 2

A linear differential equation of order $n$ with constant coefficients is of the form:

$$\frac{d^ny}{dx^n} + a_1\frac{d^{n-1}y}{dx^{n-1}} + ... + a_ny = X(x)$$

where $a_1, a_2, ..., a_n$ are constants and $X(x)$ is a function of $x$.

To simplify writing, we use the differential operator $D = \frac{d}{dx}$.

• $Dy = \frac{dy}{dx}$
• $D^2y = \frac{d^2y}{dx^2}$
• $D^ny = \frac{d^ny}{dx^n}$

The general equation becomes:
$(D^n + a_1D^{n-1} + ... + a_n)y = X$

Or simply, $f(D)y = X$.

To find the Complementary Function (C.F.), we equate $f(D)y = 0$.

We form the Auxiliary Equation (A.E.) by replacing $D$ with $m$ and equating to zero:
$$f(m) = 0$$
$$m^n + a_1m^{n-1} + ... + a_n = 0$$

The nature of the roots of this polynomial equation determines the C.F.

⇒Case 1: Roots are Real and Distinct

If the A.E. has real, unequal roots $m_1, m_2, ..., m_n$, then the C.F. is:

$$C.F. = C_1e^{m_1x} + C_2e^{m_2x} + ... + C_ne^{m_nx}$$

Example: Solve $(D^2 - 5D + 6)y = 0$

A.E.: $m^2 - 5m + 6 = 0$
$(m-2)(m-3) = 0 \Rightarrow m = 2, 3$

$C.F. = C_1e^{2x} + C_2e^{3x}$

⇒Case 2: Roots are Real and Equal (Repeated)

If a root $m_1$ is repeated twice ($m_1 = m_2 = m$), the C.F. is:

$$C.F. = (C_1 + C_2x)e^{mx}$$

If repeated three times ($m_1 = m_2 = m_3 = m$):

$$C.F. = (C_1 + C_2x + C_3x^2)e^{mx}$$

Example: $(D^2 - 4D + 4)y = 0$
A.E.: $m^2 - 4m + 4 = 0 \Rightarrow m = 2, 2$
$C.F. = (C_1 + C_2x)e^{2x}$

⇒Case 3: Roots are Complex Conjugates

If the roots are $\alpha \pm i\beta$, the C.F. is:

$$C.F. = e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$$

⇒Case 4: Repeated Complex Roots

If $\alpha \pm i\beta$ occurs twice:

$$C.F. = e^{\alpha x}[(C_1 + C_2x)\cos\beta x + (C_3 + C_4x)\sin\beta x]$$

The Particular Integral deals with the Right Hand Side $X(x)$.

$$P.I. = \frac{1}{f(D)} X$$

We use shortcut methods based on the specific function $X(x)$. We never use the general integration method unless the shortcuts fail.

⇒When $X = e^{ax}$

$$P.I. = \frac{1}{f(D)} e^{ax} = \frac{1}{f(a)} e^{ax}$$
(Replace $D$ with $a$)

⇒When $X = \sin(ax)$ or $\cos(ax)$

$$P.I. = \frac{1}{f(D^2)} \sin(ax) = \frac{1}{f(-a^2)} \sin(ax)$$
(Replace ONLY $D^2$ with $-a^2$)

Note: Do not square $(-a)$. For $\sin(3x)$, replace $D^2$ with $-9$, NOT $+9$.

If $f(-a^2) = 0$ (failure case), multiply by $x$ and differentiate the denominator: $x \frac{1}{f'(D)} \sin(ax)$.

⇒When $X = x^m$

1. 1.Write $P.I. = \frac{1}{f(D)} x^m$.
2. 2.Take the lowest degree term common from $f(D)$ to make the denominator of the form $(1 \pm \phi(D))$.
3. 3.Take it to the numerator as $(1 \pm \phi(D))^{-1}$.
4. 4.Expand using binomial theorem up to $D^m$.
5. 5.Operate on $x^m$.

⇒When $X = e^{ax} \cdot V(x)$

This is the Shift Theorem.

$$P.I. = \frac{1}{f(D)} [e^{ax} V] = e^{ax} \frac{1}{f(D+a)} V$$

Steps:
1. Pull $e^{ax}$ to the left side of the operator.
2. Replace every $D$ in the denominator with $(D+a)$.
3. Solve $\frac{1}{f(D+a)} V$ using previous rules.

⇒When $X = x \cdot V(x)$

Where $V(x)$ is usually $\sin(ax)$ or $\cos(ax)$.

$$P.I. = \left[ x - \frac{f'(D)}{f(D)} \right] \frac{1}{f(D)} V$$

This formula avoids complicated integrations.

A linear ODE where the degree of the $x$ multiplier equals the order of the derivative:

$$x^n \frac{d^ny}{dx^n} + a_1 x^{n-1} \frac{d^{n-1}y}{dx^{n-1}} + ... + a_n y = X$$

This equation has variable coefficients, so we cannot use $f(D)$ shortcuts directly. We must reduce it to constant coefficients first.

1. 1.Substitute $x = e^z$, so $z = \ln x$.
2. 2.Let $D_1 = \frac{d}{dz}$.
3. 3.Substitute:
   $x \frac{dy}{dx} = D_1 y$
   $x^2 \frac{d^2y}{dx^2} = D_1(D_1 - 1)y$
   $x^3 \frac{d^3y}{dx^3} = D_1(D_1 - 1)(D_1 - 2)y$
4. 4.The equation now becomes a Linear ODE with Constant Coefficients in terms of $z$.
5. 5.Solve for C.F. and P.I. using standard rules.
6. 6.Replace $z$ with $\ln x$ at the end.

A generalization of Cauchy's equation:

$$(ax+b)^n \frac{d^ny}{dx^n} + c_1(ax+b)^{n-1} \frac{d^{n-1}y}{dx^{n-1}} + ... = X$$

1. 1.Substitute $ax+b = e^z$, so $z = \ln(ax+b)$.
2. 2.Let $D_1 = \frac{d}{dz}$.
3. 3.Substitute:
   $(ax+b) \frac{dy}{dx} = a D_1 y$
   $(ax+b)^2 \frac{d^2y}{dx^2} = a^2 D_1(D_1 - 1)y$
4. 4.Solve the resulting constant-coefficient equation.

A powerful method to find the P.I. when the shortcut rules fail (e.g., $X = \tan x, \sec x, \ln x$).

For a second-order ODE: $\frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy = X$

1. 1.Find the C.F.: $y_c = C_1 y_1 + C_2 y_2$.
2. 2.Calculate the Wronskian: $W = y_1 y_2' - y_2 y_1'$. (Ensure $W \neq 0$).
3. 3.The P.I. is given by $u \cdot y_1 + v \cdot y_2$, where:
   $u = -\int \frac{y_2 \cdot X}{W} dx$
   $v = \int \frac{y_1 \cdot X}{W} dx$

Systems with one independent variable ($t$) and two dependent variables ($x, y$).

Example:
$\frac{dx}{dt} + 2y = e^t$
$\frac{dy}{dt} - x = \sin t$

⇒Method:

Use the operator $D = \frac{d}{dt}$. Treat the equations like simultaneous algebraic equations to eliminate one dependent variable, solve the resulting higher-order ODE for the other, and back-substitute.

Higher order ODEs model Series L-C-R circuits.

By Kirchhoff's Voltage Law:
$$L\frac{di}{dt} + Ri + \frac{1}{C}q = E(t)$$

Since $i = \frac{dq}{dt}$ and $\frac{di}{dt} = \frac{d^2q}{dt^2}$, we get a 2nd order ODE:
$$L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = E(t)$$

The C.F. represents the transient current (which dies out over time), and the P.I. represents the steady-state current.

ODEs also model a mass attached to a spring.

$$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)$$

• $m$ = mass
• $c$ = damping coefficient (friction/air resistance)
• $k$ = spring constant (stiffness)
• $F(t)$ = external driving force

The nature of the A.E. roots determines if the system is Overdamped (real roots), Critically Damped (equal roots), or Underdamped (complex roots - oscillating).

⇒Before the exam, ensure you can:

• Write the C.F. for repeated complex roots.
• State the P.I. rule for $\sin(ax)$ and what to do if $f(-a^2) = 0$.
• Apply the shift theorem $e^{ax}V$.
• Convert a Cauchy-Euler equation into a constant coefficient equation using $x=e^z$.
• Write the formula for the Wronskian in Variation of Parameters.
• Explain the physical meaning of C.F. and P.I. in an LCR electrical circuit.