Differential Equations – Solving and Applications

Introduction

A differential equation (DE) is an equation that contains an unknown function and its derivatives.
The order of a DE is the highest derivative present.
A DE is linear if the unknown function and its derivatives appear to the first power and are not multiplied together.
A DE is homogeneous if it contains only terms involving the unknown function and its derivatives; otherwise it is non‑homogeneous.

The general solution of a DE is the sum of the complementary (homogeneous) solution and a particular solution.
Initial or boundary conditions are used to determine the constants that appear in the general solution.

First‑Order Differential Equations

1. Separable Equations

A DE of the form
$$\frac{dy}{dx}=g(x)h(y)$$
is separable.
Procedure

1. Rewrite as $\frac{1}{h(y)}dy=g(x)dx$.
2. Integrate both sides.
3. Solve for $y$.

Worked Example – Population Growth
The user base of a mobile app grows proportionally to its current size:
$$\frac{dU}{dt}=kU,U(0)=U_0$$
with $k=0.02$ day${}^{-1}$ and $U_0=1000$.

$$\frac{1}{U}dU=0.02dt⟹\ln U=0.02t+C$$
Exponentiating and applying $U(0)=1000$:
$$U(t)=1000e^{0.02t}$$

The graph of $U(t)$ is an exponential curve.

2. Linear Equations

A DE of the form
$$\frac{dy}{dx}+P(x)y=Q(x)$$
is linear.
Integrating Factor
$$\mu (x)=e^{\int P(x)dx}$$
Multiplying the DE by $\mu (x)$ gives
$$\frac{d}{dx}[\mu (x)y]=\mu (x)Q(x)$$
Integrate and solve for $y$.

Mermaid Diagram – Solving a Linear DE

flowchart TD
  A["Start: \(\frac{dy}{dx}+P(x)y=Q(x)\)"] --> B["Compute \(\mu(x)=e^{\int P(x)dx}\)"]
  B --> C["Multiply DE by \(\mu(x)\)"]
  C --> D["Recognize \(\frac{d}{dx}[\mu y]\)"]
  D --> E["Integrate: \(\mu y=\int \mu Q dx + C\)"]
  E --> F["Solve for \(y\)"]
  F --> G["Apply initial condition"]
  G --> H["Final solution"]

Worked Example – RC Circuit
The voltage $V(t)$ across a capacitor in an RC circuit satisfies
$$\frac{dV}{dt}+\frac{1}{RC}V=\frac{E}{RC}$$
with $R=10\text{k}\Omega$, $C=100\mu \text{F}$, $E=5\text{V}$, and $V(0)=0$.

$$\mu (t)=e^{t/(RC)}=e^{t/1}(\text{since }RC=1\text{s})$$
$$\frac{d}{dt}\left[e^{t}V\right]=\frac{5}{1}{e}^t$$
Integrate:
$$e^{t}V=5e^t+C⟹V(t)=5+C{e}^{-t}$$
Using $V(0)=0$: $C=-5$.
$$V(t)=5(1-e^{-t})\text{V}$$

The charging curve is shown below.

Real‑world Picture

3. Exact Equations

A DE $M(x,y)dx+N(x,y)dy=0$ is exact if
$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$
Then there exists a potential function $\psi (x,y)$ such that
$$\frac{\partial \psi }{\partial x}=M,\frac{\partial \psi }{\partial y}=N$$
Integrate $M$ with respect to $x$ and add an arbitrary function of $y$, then differentiate with respect to $y$ to find that function.

4. Homogeneous and Bernoulli Equations

A homogeneous first‑order DE has the form
$$\frac{dy}{dx}=F\left(\frac{y}{x}\right)$$
Substitute $v=y/x$.
A Bernoulli equation
$$\frac{dy}{dx}+P(x)y=Q(x)y^n$$
is linear in $u=y^{1-n}$.

Second‑Order Linear Differential Equations

1. Homogeneous with Constant Coefficients

The general form
$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$$
has characteristic equation
$$a{r}^2+br+c=0$$
Solve for $r$.

• Distinct real roots $r_1,r_2$:
  $$y_c=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$$
• Repeated real root $r$:
  $$y_c=(C_1+C_{2}x)e^{rx}$$
• Complex conjugate roots $\alpha \pm i\beta$:
  $$y_c=e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x)$$

Mermaid Diagram – Solving a Second‑Order Linear DE

flowchart TD
  A["Start: \(ay''+by'+cy=0\)"] --> B["Form characteristic \(ar^2+br+c=0\)"]
  B --> C["Solve for roots \(r_1,r_2\)"]
  C --> D["Case analysis"]
  D --> E1["Distinct real: \(y_c=C_1e^{r_1x}+C_2e^{r_2x}\)"]
  D --> E2["Repeated: \(y_c=(C_1+C_2x)e^{rx}\)"]
  D --> E3["Complex: \(y_c=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)\)"]
  E1 & E2 & E3 --> F["Add particular solution if non‑homogeneous"]
  F --> G["Apply initial/boundary conditions"]
  G --> H["Final solution"]

Worked Example – Damped Oscillator
Solve
$$y^{\prime \prime }+3y^{\prime }+2y=0$$
Characteristic equation:
$$r^2+3r+2=0⟹(r+1)(r+2)=0$$
Roots: $r_1=-1,r_2=-2$.
Complementary solution:
$$y_c=C_1{e}^{-x}+C_2{e}^{-2x}$$

If the system is forced by a sinusoid $f(t)=\sin t$, the non‑homogeneous equation
$$y^{\prime \prime }+3y^{\prime }+2y=\sin t$$
has a particular solution $y_p=A\sin t+B\cos t$.
Substituting and solving for $A,B$ gives $A=-\frac{1}{10},B=\frac{3}{10}$.
Thus
$$y(t)=C_1{e}^{-t}+C_2{e}^{-2t}-\frac{1}{10}\sin t+\frac{3}{10}\cos t$$

The graph of the damped oscillation is shown below.

2. Non‑Homogeneous Equations

Two common methods:

• Method of Undetermined Coefficients – guess a particular solution based on the form of $f(x)$.
• Variation of Parameters – use the complementary solution to construct a particular solution.

Worked Example – Logistic Growth
The logistic model for a population $P(t)$ with carrying capacity $K$ and intrinsic growth rate $r$ is
$$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$$
This is a separable first‑order DE.
$$\frac{dP}{P(1-P/K)}=rdt$$
Partial fractions give
$$\frac{1}{P}+\frac{1}{K-P}=\frac{r}{K}dt$$
Integrate:
$$\ln \left(\frac{P}{K-P}\right)=\frac{r}{K}t+C$$
Solve for $P(t)$:
$$P(t)=\frac{K}{1+A{e}^{-rt}},A=\frac{K-P_0}{P_0}$$

If $K=10000$ users, $r=0.05$ day${}^{-1}$, and $P_0=500$, then
$$A=\frac{10000-500}{500}=19$$
$$P(t)=\frac{10000}{1+19e^{-0.05t}}$$

The logistic curve is plotted below.

Comparison of First‑Order Methods

In the Real World

1. Mobile Phone Battery Charging (Ncell, NTC)
   The voltage across a capacitor in the charging circuit satisfies
   $$\frac{dV}{dt}+\frac{1}{RC}V=\frac{E}{RC}$$
   The solution $V(t)=E(1-e^{-t/RC})$ predicts how quickly the battery reaches full charge.

2. Daraz Order Queue (E‑commerce)
   Let $L(t)$ be the average number of orders in the queue.
   $$\frac{dL}{dt}=\lambda -\mu L$$
   where $\lambda$ is the arrival rate and $\mu$ the service rate.
   Solving gives $L(t)=\frac{\lambda }{\mu }+C{e}^{-\mu t}$, showing how the queue stabilises.

3. NEPSE Stock Price Trend (Finance)
   A simple model for the rate of change of a stock price $S(t)$ is
   $$\frac{dS}{dt}=kS$$
   with $k$ reflecting market sentiment.
   The exponential solution $S(t)=S_0{e}^{kt}$ captures rapid growth or decay observed in volatile markets.

Exam Tip

• Identify the type of DE first: separable, linear, exact, homogeneous, Bernoulli, or second‑order linear.
• Write down the integrating factor immediately for linear equations.
• For second‑order constant‑coefficient equations, always solve the characteristic equation and check the nature of its roots.
• Remember the structure of the general solution: complementary part + particular part.
• Use initial conditions to find constants; show each substitution step clearly.
• Show the graph of the solution if the question asks for a visual interpretation.
• Practice converting real‑world rates (e.g., growth, decay, charging) into differential equations before solving.

Good luck!