Ordinary Differential Equations (ODEs): Types, Solutions & Applications

TAKEAWAYS:

• ODEs model real-world rates of change: From population growth (e.g., Nepal’s demographic projections) to cooling coffee (Newton’s Law of Cooling) and drug concentration in blood.
• First-order ODEs solve separable, linear, and exact equations using substitution, integrating factors, or exactness conditions—each with a distinct method and verification step.
• Second-order ODEs with constant coefficients require characteristic equations, roots (real/distinct, real/repeated, complex), and general solutions built from eigenfunctions.
• Initial/boundary conditions turn general solutions into unique particular solutions (e.g., determining a specific loan repayment schedule or bridge deflection).
• Applications span physics (spring-mass systems), biology (predator-prey models), and finance (interest compounding)—always tie theory to concrete examples.
• Exam focus: Solve ODEs step-by-step, classify types, and interpret real-world scenarios (e.g., "A tank’s salt concentration over time" or "Pathao’s rider demand growth").

1. What Are Ordinary Differential Equations (ODEs)?

ODEs describe how a single variable (e.g., temperature $T$, population $P$, voltage $V$) changes with respect to one independent variable (usually time $t$ or position $x$). They are fundamental to modeling dynamic systems.

Classification by Order and Linearity

classDiagram
    class ODE {
        <<abstract>>
        +order: integer
        +linearity: boolean
    }
    class FirstOrder {
        +form: dy/dx = f(x,y)
        +methods: Separable, Linear, Exact
    }
    class SecondOrder {
        +form: d²y/dx² + p(x)dy/dx + q(x)y = g(x)
        +methods: Constant Coefficients, Reduction of Order
    }
    ODE <|-- FirstOrder
    ODE <|-- SecondOrder

Key Definitions:

• Order: Highest derivative in the equation (e.g., $d^{2}y/d{x}^2$ → 2nd order).
• Linearity: An ODE is linear if:
  • The dependent variable $y$ and all its derivatives appear linearly (no $y^2$, $\sin (y)$, etc.).
  • Coefficients depend only on the independent variable (e.g., $p(x)$, $q(x)$).
• Homogeneous: $g(x)=0$ (no forcing term). Non-homogeneous: $g(x)\ne 0$.

2. First-Order ODEs: Solving Methods

First-order ODEs have the form: $$\frac{dy}{dx}=f(x,y)$$ We solve them using three primary methods, each with a unique approach.

A. Separable Equations

Form: $dy/dx=g(x)h(y)$ Method: Rewrite as $\frac{dy}{h(y)}=g(x)dx$, then integrate both sides.

Worked Example 1: Population Growth (Nepal’s Demographic Model) Suppose Nepal’s population $P(t)$ grows at a rate proportional to its current size: $$\frac{dP}{dt}=0.02P,P(0)=30\text{ million}$$ Solution:

1. Separate variables: $$\frac{dP}{P}=0.02dt$$
2. Integrate: $$\ln |P|=0.02t+C$$
3. Exponentiate: $$P(t)=C{e}^{0.02t}$$
4. Apply initial condition $P(0)=30$: $$30=C{e}^0\Rightarrow C=30$$
5. Final solution: $$P(t)=30e^{0.02t}$$ Visualization: Real-World Tie: This models Nepal’s population growth (2% annual growth rate). For $t=20$ years, $P(20)\approx 30e^{0.4}\approx 40.9$ million.

B. Linear First-Order ODEs

Form: $dy/dx+P(x)y=Q(x)$ Method: Use an integrating factor $\mu (x)=e^{\int P(x)dx}$.

Worked Example 2: Newton’s Law of Cooling (Khalti’s Server Temperature) A Khalti server’s temperature $T(t)$ cools in a room at $20^{\circ }C$. The cooling rate is proportional to the temperature difference: $$\frac{dT}{dt}=-0.1(T-20),T(0)=80^{\circ }C$$ Solution:

1. Rewrite in standard form: $$\frac{dT}{dt}+0.1T=2$$
2. Identify $P(x)=0.1$, $Q(x)=2$.
3. Compute integrating factor: $$\mu (t)=e^{\int 0.1dt}=e^{0.1t}$$
4. Multiply through by $\mu (t)$: $$e^{0.1t}\frac{dT}{dt}+0.1e^{0.1t}T=2e^{0.1t}$$ Left side is $\frac{d}{dt}(e^{0.1t}T)$.
5. Integrate: $$e^{0.1t}T=20e^{0.1t}+C$$
6. Solve for $T(t)$: $$T(t)=20+C{e}^{-0.1t}$$
7. Apply $T(0)=80$: $$80=20+C\Rightarrow C=60$$
8. Final solution: $$T(t)=20+60e^{-0.1t}$$ Visualization: Real-World Tie: This models how a Daraz server’s CPU temperature drops after shutdown. At $t=20$ minutes, $T(20)\approx 20+60e^{-2}\approx 38.5^{\circ }C$.

C. Exact Equations

Form: $M(x,y)dx+N(x,y)dy=0$ is exact if $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. Method: Find potential function $\psi (x,y)$ such that $\frac{\partial \psi }{\partial x}=M$ and $\frac{\partial \psi }{\partial y}=N$.

Worked Example 3: Work Done by a Variable Force (Pathao Rider’s Effort) A Pathao rider’s effort $F$ depends on speed $v$ and distance $x$: $$F=(2x+3y)dx+(4x+6y)dy$$ Check exactness: $$\frac{\partial M}{\partial y}=3,\frac{\partial N}{\partial x}=4$$ Since $3\ne 4$, the equation is not exact. However, if we adjust $F$ to: $$F=(2x+3y)dx+(3x+6y)dy$$ Now $\frac{\partial M}{\partial y}=3=\frac{\partial N}{\partial x}$, so it’s exact. Solution:

1. Integrate $M$ w.r.t. $x$: $$\psi (x,y)=\int (2x+3y)dx=x^2+3xy+h(y)$$
2. Differentiate $\psi$ w.r.t. $y$ and set equal to $N$: $$\frac{\partial \psi }{\partial y}=3x+h^{\prime }(y)=3x+6y\Rightarrow h^{\prime }(y)=6y$$
3. Integrate $h^{\prime }(y)$: $$h(y)=3y^2+C$$
4. General solution: $$\psi (x,y)=x^2+3xy+3y^2=C$$ Visualization:

3. Second-Order Linear ODEs with Constant Coefficients

Form: $$a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=g(x)$$ Method:

1. Solve the homogeneous equation ($g(x)=0$) using the characteristic equation.
2. Find a particular solution for $g(x)\ne 0$ (e.g., polynomial, exponential, trigonometric).
3. Combine solutions.

A. Homogeneous Solutions

The characteristic equation is: $$a{r}^2+br+c=0$$ Roots and Solutions:

Worked Example 4: Spring-Mass System (NTC’s Power Grid Oscillation) A mass $m=2$ kg on a spring with $k=8$ N/m and damping $c=4$ N·s/m satisfies: $$2\frac{d^{2}x}{d{t}^2}+4\frac{dx}{dt}+8x=0$$ Solution:

1. Characteristic equation: $$2r^2+4r+8=0\Rightarrow r^2+2r+4=0$$
2. Roots: $$r=\frac{-2\pm \sqrt{4-16}}{2}=-1\pm i\sqrt{3}$$
3. General solution: $$x(t)=e^{-t}(C_1\cos (\sqrt{3}t)+C_2\sin (\sqrt{3}t))$$ Visualization:

B. Particular Solutions (Non-Homogeneous Case)

Use method of undetermined coefficients for $g(x)$ of the form:

• Polynomial: $P_n(x)$
• Exponential: $e^{ax}$
• Trigonometric: $\sin (bx)$ or $\cos (bx)$

Worked Example 5: RC Circuit (Ncell’s Charging Battery) A battery charges a capacitor with voltage $V(t)$ satisfying: $$\frac{d^{2}V}{d{t}^2}+5\frac{dV}{dt}+6V=12$$ Solution:

1. Homogeneous solution: Characteristic equation: $r^2+5r+6=0\Rightarrow r=-2,-3$ $$V_h(t)=C_1{e}^{-2t}+C_2{e}^{-3t}$$
2. Particular solution (guess $V_p=A$): $$6A=12\Rightarrow A=2$$
3. General solution: $$V(t)=C_1{e}^{-2t}+C_2{e}^{-3t}+2$$ Visualization:

4. Applications of ODEs in Real World

ODEs model dynamic systems where change depends on current state. Here’s how Nepalese and global companies use them:

Worked Example 6: Loan Amortization (Bank Loan Interest) A bank offers a loan with monthly payments $P$ and interest rate $r=0.005$ (0.5% per month). The remaining balance $B(t)$ satisfies: $$\frac{dB}{dt}=-P+rB,B(0)=100,000$$ Solution:

1. Rewrite: $$\frac{dB}{dt}+0.005B=-P$$
2. Integrating factor: $$\mu (t)=e^{0.005t}$$
3. Multiply and integrate: $$B(t)=C{e}^{-0.005t}+\frac{P}{0.005}$$
4. Apply $B(0)=100,000$ and $B(T)=0$ (loan paid off at time $T$): $$100,000=C+\frac{P}{0.005},0=C{e}^{-0.005T}+\frac{P}{0.005}$$
5. Solve for $P$: $$P=\frac{0.005\times 100,000}{1-e^{-0.005T}}$$ For $T=120$ months (10 years), $P\approx 955.22$ NPR/month.

5. Exam Tips for ODEs

1. Classify the ODE first: Is it first-order separable, linear, or exact? Or second-order with constant coefficients?
2. Check exactness carefully: $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ must hold for exact equations.
3. For second-order ODEs:
   • Always write the characteristic equation.
   • Handle repeated/complex roots correctly.
   • For non-homogeneous terms, guess a particular solution based on $g(x)$.
4. Initial conditions are crucial: They determine constants $C_1,C_2,\dots$. Never skip applying them!
5. Real-world interpretation: Exams often ask to explain the physical meaning (e.g., "What does $C_1$ represent in the population model?").
6. Common mistakes to avoid:
   • Forgetting the integrating factor in linear ODEs.
   • Misapplying initial conditions (e.g., using $t=0$ incorrectly).
   • Skipping the homogeneous solution for non-homogeneous ODEs.

6. Practice Problems (Exam-Style)

1. Separable ODE: Solve $\frac{dy}{dx}=x{y}^2$ with $y(0)=1$. Hint: Separate and integrate.

2. Linear ODE: Solve $\frac{dy}{dx}+3y=e^{-2x}$ with $y(0)=0$.

3. Exact ODE: Verify and solve $(2xy+y^2)dx+(x^2+2xy)dy=0$.

4. Second-Order ODE: Solve $\frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}+4y=0$ with $y(0)=1$, $y^{\prime }(0)=0$.

5. Application: A tank contains 1000 L of brine with 50 kg of salt. Pure water enters at 5 L/min, and the mixture drains at 5 L/min. Find the salt amount $Q(t)$ over time. Hint: Use a first-order linear ODE based on rate of change.

7. Summary Table of Solution Methods

8. Final Visual: ODE Solution Roadmap