Differential Equations: Types, Solutions & Applications

TAKEAWAYS:

• Differential equations model real-world rates of change (e.g., population growth, loan interest, cooling objects).
• First-order equations are solved by separation of variables, integrating factors, or substitution; second-order equations require characteristic equations or reduction of order.
• Homogeneous equations use substitution $y=vx$, while linear equations use integrating factors $e^{\int P(x)dx}$.
• Exact equations require $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$, solved by integrating $M$ and $N$.
• Applications include compound interest (exponential growth), cooling laws (Newton’s law), and spring-mass systems (harmonic motion).
• Numerical methods (Euler’s method) approximate solutions when analytical methods fail.

1. Introduction to Differential Equations

A differential equation (DE) is an equation involving an unknown function and its derivatives. It describes how a quantity changes with respect to another (e.g., time, distance).

Types of Differential Equations

Worked Example 1: Separation of Variables

Problem: Solve $\frac{dy}{dx}=xy$ with $y(0)=1$. Solution:

1. Separate variables: $\frac{dy}{y}=xdx$.
2. Integrate both sides: $$\int \frac{dy}{y}=\int xdx⟹\ln |y|=\frac{x^2}{2}+C.$$
3. Exponentiate: $y=e^{\frac{x^2}{2}+C}=C{e}^{\frac{x^2}{2}}$.
4. Apply initial condition $y(0)=1$: $$1=C{e}^0⟹C=1.$$ Final Solution: $y=e^{\frac{x^2}{2}}$.

Worked Example 2: Integrating Factor (Linear DE)

Problem: Solve $\frac{dy}{dx}+2y=e^{-x}$. Solution:

1. Identify $P(x)=2$, $Q(x)=e^{-x}$.
2. Integrating factor $\mu (x)=e^{\int P(x)dx}=e^{2x}$.
3. Multiply through: $e^{2x}\frac{dy}{dx}+2e^{2x}y=1$.
4. Left side is $\frac{d}{dx}(e^{2x}y)$, so integrate: $$e^{2x}y=\int 1dx=x+C.$$
5. Solve for $y$: $$y=e^{-2x}(x+C).$$

2. Homogeneous Differential Equations

A first-order DE $\frac{dy}{dx}=f(x,y)$ is homogeneous if $f(x,y)$ can be written as $g\left(\frac{y}{x}\right)$. Solve by substitution $y=vx$, so $\frac{dy}{dx}=v+x\frac{dv}{dx}$.

Worked Example 3: Homogeneous Equation

Problem: Solve $\frac{dy}{dx}=\frac{x^2+y^2}{xy}$. Solution:

1. Substitute $y=vx$, $\frac{dy}{dx}=v+x\frac{dv}{dx}$: $$v+x\frac{dv}{dx}=\frac{x^2+v^2{x}^2}{x(vx)}=\frac{1+v^2}{v}.$$
2. Separate variables: $$x\frac{dv}{dx}=\frac{1+v^2}{v}-v=\frac{1}{v}.$$
3. Integrate: $$\int vdv=\int \frac{1}{x}dx⟹\frac{v^2}{2}=\ln |x|+C.$$
4. Substitute back $v=\frac{y}{x}$: $$\frac{y^2}{2x^2}=\ln |x|+C⟹y^2=2x^2(\ln |x|+C).$$

3. Exact Differential Equations

An equation $M(x,y)dx+N(x,y)dy=0$ is exact if: $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.$$ Solution: Integrate $M$ w.r.t. $x$ and $N$ w.r.t. $y$, then combine.

Worked Example 4: Exact Equation

Problem: Solve $(2xy+3)dx+(x^2+4y)dy=0$. Solution:

1. Check exactness: $$\frac{\partial M}{\partial y}=2x,\frac{\partial N}{\partial x}=2x.$$ Exact.
2. Integrate $M$ w.r.t. $x$: $$\int (2xy+3)dx=x^{2}y+3x+h(y).$$
3. Differentiate w.r.t. $y$ and compare with $N$: $$\frac{\partial }{\partial y}(x^{2}y+3x+h(y))=x^2+h^{\prime }(y)=x^2+4y⟹h^{\prime }(y)=4y.$$
4. Integrate $h^{\prime }(y)$: $$h(y)=2y^2+C.$$
5. General solution: $$x^{2}y+3x+2y^2=C.$$

4. Second-Order Linear Differential Equations

Form: $a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=0$. Solve using characteristic equation $a{r}^2+br+c=0$.

Cases:

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

Worked Example 5: Characteristic Equation

Problem: Solve $\frac{d^{2}y}{d{x}^2}-3\frac{dy}{dx}+2y=0$. Solution:

1. Characteristic equation: $$r^2-3r+2=0⟹(r-1)(r-2)=0⟹r=1,2.$$
2. General solution: $$y=C_1{e}^x+C_2{e}^{2x}.$$

Worked Example 6: Damped Harmonic Motion (Real-World Application)

Problem: A spring-mass system has equation $\frac{d^{2}x}{d{t}^2}+4\frac{dx}{dt}+4x=0$. Find displacement $x(t)$. Solution:

1. Characteristic equation: $$r^2+4r+4=0⟹(r+2{)}^2=0⟹r=-2\text{ (repeated)}.$$
2. General solution: $$x(t)=(C_1+C_{2}t)e^{-2t}.$$ Interpretation: The mass oscillates with exponentially decaying amplitude due to damping.

5. Applications of Differential Equations

In the Real World

1. eSewa/Khalti (Loan Interest Calculation):

   • Idea: Exponential Growth (First-Order DE)
   • How: Compound interest is modeled by $\frac{dA}{dt}=rA$, where $A$ is principal, $r$ is interest rate.
   • Example: A loan of Rs. 100,000 at 10% annual interest grows as $A(t)=100000e^{0.1t}$. After 5 years: $$A(5)=100000e^{0.5}\approx 164,872.$$

2. Pathao (Traffic Flow Modeling):

   • Idea: First-Order Linear DE (Traffic Density)
   • How: Traffic flow $q$ relates to density $\rho$ via $\frac{d\rho }{dt}=-\frac{\partial q}{\partial x}$. Pathao’s algorithm optimizes routes by solving DEs for congestion prediction.

3. NTC (Newton’s Law of Cooling):

   • Idea: First-Order Linear DE
   • How: Temperature $T(t)$ of a cooling object satisfies $\frac{dT}{dt}=-k(T-T_{\text{env}})$, where $k$ is cooling constant.
   • Example: A cup of tea at 90°C in a 25°C room cools to 50°C in 10 minutes. Solve: $$\frac{dT}{dt}=-k(T-25).$$ Separate variables and integrate to find $k$, then predict time to reach 40°C.

4. NEPSE (Stock Price Trends):

   • Idea: Second-Order DE (Market Dynamics)
   • How: Stock prices $P(t)$ may follow $\frac{d^{2}P}{d{t}^2}+a\frac{dP}{dt}+bP=f(t)$, where $f(t)$ is external demand.
   • Example: If $\frac{d^{2}P}{d{t}^2}+0.5\frac{dP}{dt}+P=100$, solve the homogeneous part first, then find particular solution.

6. Numerical Methods: Euler’s Method

When analytical solutions are difficult, use Euler’s method to approximate solutions: $$y_{n+1}=y_n+hf(x_n,y_n),$$ where $h$ is step size.

Worked Example 7: Euler’s Method

Problem: Approximate $y(0.2)$ for $\frac{dy}{dx}=x+y$, $y(0)=1$, with $h=0.1$. Solution:

1. Iterate:
   • $x_0=0$, $y_0=1$.
   • $y_1=y_0+h(x_0+y_0)=1+0.1(0+1)=1.1$.
   • $y_2=y_1+h(x_1+y_1)=1.1+0.1(0.1+1.1)=1.22$.
2. Approximate $y(0.2)\approx 1.22$.

Exam Tip

1. Classify the DE first: Identify order, linearity, homogeneity, or exactness before solving.
2. Check initial conditions: Always apply them to find constants $C$.
3. Graph solutions: Sketch graphs for first-order DEs (e.g., exponential growth/decay) and second-order DEs (oscillatory vs. exponential).
4. Common mistakes:
   • Forgetting to multiply by the integrating factor.
   • Misapplying substitution for homogeneous equations.
   • Incorrectly solving characteristic equations (factor quadratics properly).
5. Application questions: Relate DEs to real-world scenarios (e.g., "A tank drains water..." or "Population grows..."). Always define variables clearly (e.g., $y(t)=$ population at time $t$).

Visual Summary of Solution Methods:

mindmap
  root((Differential Equations))
    First-Order
      Separation of Variables
      Integrating Factor
      Homogeneous Substitution
      Exact Equations
    Second-Order
      Characteristic Equation
      Repeated Roots
      Complex Roots
    Applications
      Finance (Interest)
      Physics (Spring-Mass)
      Biology (Population)
    Numerical Methods
      Euler's Method