Differential Equations: Types, Solutions & Applications

TAKEAWAYS:

• Differential equations relate a function to its derivatives and model real-world dynamic systems.
• First-order equations can be solved by separation of variables, integrating factors, or substitution.
• Second-order linear equations require characteristic equations and complementary/particular solutions.
• Applications include population growth, cooling laws, and spring-mass systems.
• Graphical solutions (slope fields) help visualize solutions when analytical methods fail.
• Always check initial conditions to find specific solutions from general ones.

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What is a Differential Equation?

A differential equation is an equation that involves an unknown function and its derivatives. It describes how a quantity changes over time or space.

Example: The equation $\frac{dy}{dx}=2x$ is a differential equation because it relates the derivative of $y$ with respect to $x$ to $x$ itself.

Why do we study them? Differential equations model real-world phenomena like:

• Population growth (biology)
• Heat flow (physics)
• Economic trends (finance)

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Types of Differential Equations

Differential equations are classified based on:

1. Order: The highest derivative in the equation.

   • First-order: $\frac{dy}{dx}+y=0$
   • Second-order: $\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}+2y=0$

2. Linearity: Whether the equation is linear or nonlinear.

   • Linear: $\frac{dy}{dx}+P(x)y=Q(x)$
   • Nonlinear: $\frac{dy}{dx}=y^2+x$

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First-Order Differential Equations

1. Separation of Variables

Method: Rewrite the equation so that all $y$-terms are on one side and $x$-terms on the other.

Example: Solve $\frac{dy}{dx}=ky$, where $k$ is a constant.

Solution:

1. Separate variables: $\frac{dy}{y}=kdx$
2. Integrate both sides: $\int \frac{dy}{y}=\int kdx$ $\ln |y|=kx+C$
3. Solve for $y$: $y=e^{kx+C}=C{e}^{kx}$ (where $C=e^C$).

Graphical Interpretation:

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2. Integrating Factor Method

Use: For linear first-order equations of the form: $\frac{dy}{dx}+P(x)y=Q(x)$.

Steps:

1. Find the integrating factor $\mu (x)=e^{\int P(x)dx}$.
2. Multiply both sides of the equation by $\mu (x)$.
3. The left side becomes the derivative of $y\cdot \mu (x)$.
4. Integrate and solve for $y$.

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

Solution:

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

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3. Homogeneous Equations

Form: $\frac{dy}{dx}=f\left(\frac{y}{x}\right)$.

Method: Substitute $y=vx$, then solve for $v$.

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

Solution:

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

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Second-Order Differential Equations

1. Linear Equations with Constant Coefficients

Form: $a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=0$.

Method: Solve the 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: $y=e^{\alpha x}(C_1\cos (\beta x)+C_2\sin (\beta x))$.

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

Solution:

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

Graphical Interpretation:

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2. Nonhomogeneous Equations

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

Method: Find the complementary solution $y_c$ and a particular solution $y_p$. The general solution is $y=y_c+y_p$.

Example: Solve $y^{\prime \prime }-y=e^x$.

Solution:

1. Complementary solution: $y_c=C_1{e}^x+C_2{e}^{-x}$.
2. Guess $y_p=Ax{e}^x$ (since $e^x$ is already in $y_c$).
3. Compute derivatives and substitute into the equation to find $A$.
4. Final solution: $y=C_1{e}^x+C_2{e}^{-x}+\frac{x{e}^x}{2}$.

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Applications of Differential Equations

1. Population Growth (First-order): $\frac{dP}{dt}=kP$ models exponential growth. Solution: $P(t)=P_0{e}^{kt}$.

2. Newton’s Law of Cooling (First-order): $\frac{dT}{dt}=-k(T-T_{\text{env}})$. Solution describes how an object cools over time.

3. Spring-Mass Systems (Second-order): $m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$ models harmonic motion.

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Solving Differential Equations Graphically

When analytical solutions are difficult, we can use slope fields to visualize solutions.

Example: For $\frac{dy}{dx}=x^2-y^2$, draw small line segments with slope $x^2-y^2$ at grid points. Solutions are curves tangent to these segments.

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Exam Tip

1. Classify the Equation: Always identify the type (first-order, second-order, linear, nonlinear) before solving.

2. Check Initial Conditions: If initial conditions (e.g., $y(0)=5$) are given, use them to find specific constants.

3. Practice Common Forms: Memorize standard forms like separation of variables, integrating factors, and homogeneous equations.

4. Graphical Solutions: For complex equations, sketch slope fields to understand behavior.

5. Applications: Questions often link differential equations to real-world scenarios (e.g., "A tank fills with water..."). Translate the scenario into an equation first.

6. Common Mistakes:

   • Forgetting to include the constant of integration ($+C$).
   • Incorrectly applying the integrating factor.
   • Misidentifying homogeneous equations.

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NEB Board-Style Questions

Short Answer (5 marks each)

1. Solve the differential equation $\frac{dy}{dx}=\frac{y}{x}+\frac{x}{y}$ using substitution.
2. Find the general solution of $\frac{d^{2}y}{d{x}^2}+4y=0$.
3. Explain the integrating factor method with an example.
4. A population grows according to $\frac{dP}{dt}=0.02P$. If $P(0)=1000$, find $P(t)$.
5. Sketch the slope field for $\frac{dy}{dx}=x-y$ and draw two solution curves.

Long Answer (10 marks each)

1. Solve the differential equation $x\frac{dy}{dx}+y=x^2$ using the integrating factor method. Verify your solution by substitution.
2. A cooling object satisfies $\frac{dT}{dt}=-k(T-20)$, where $T$ is temperature and $20^{\circ }\text{C}$ is room temperature. If $T(0)=100^{\circ }\text{C}$ and $T(5)=60^{\circ }\text{C}$, find $k$ and $T(t)$.
3. Solve $y^{\prime \prime }+y^{\prime }-6y=e^{2x}$ using the method of undetermined coefficients.
4. Explain how differential equations model projectile motion. Derive the equation for the height $h(t)$ of a projectile under gravity, ignoring air resistance.
5. Compare and contrast separation of variables and the integrating factor method for solving first-order differential equations. Provide an example for each.

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Summary Table