First-Order Differential Equations: Types, Solutions & Applications

1. Introduction to First-Order Differential Equations (ODEs)

A first-order ODE is an equation of the form: $$\frac{dy}{dx}=f(x,y)\text{or}F(x,y,y^{\prime })=0$$ where $y$ is a function of $x$, and $y^{\prime }=\frac{dy}{dx}$. The order is the highest derivative present (here, 1). Solutions are functions $y(x)$ that satisfy the equation.

Key Concepts

• General Solution: Contains arbitrary constants (e.g., $y=C{e}^x$).
• Particular Solution: Satisfies an initial condition (e.g., $y(0)=2$).
• Singular Solution: A solution not obtainable from the general solution (rare in first-order ODEs).

---

2. Classification of First-Order ODEs

First-order ODEs are classified based on their structure. The choice of method depends on their form:

---

3. Solving Methods with Worked Examples

3.1 Separable Equations

Definition: Equations where variables can be separated as $f(y)dy=g(x)dx$.

Steps:

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

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

Solution:

1. Separate: $\frac{dy}{y}=xdx$.
2. Integrate: $\ln |y|=\frac{x^2}{2}+C$.
3. Exponentiate: $y=C{e}^{x^2/2}$.
4. Apply IC: $1=C{e}^0\Rightarrow C=1$. Final Answer: $y=e^{x^2/2}$.

Verification: Differentiate $y=e^{x^2/2}$: $$\frac{dy}{dx}=e^{x^2/2}\cdot x=xy\text{(matches the ODE)}.$$

---

3.2 Linear First-Order ODEs

Standard Form: $$\frac{dy}{dx}+P(x)y=Q(x).$$ Integrating Factor (IF): $\mu (x)=e^{\int P(x)dx}$.

Steps:

1. Compute $\mu (x)$.
2. Multiply ODE by $\mu (x)$: $\mu (x)\frac{dy}{dx}+\mu (x)P(x)y=\mu (x)Q(x)$.
3. Left side becomes $\frac{d}{dx}(\mu (x)y)$.
4. Integrate: $\mu (x)y=\int \mu (x)Q(x)dx+C$.
5. Solve for $y$.

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

Solution:

1. $P(x)=2$, so $\mu (x)=e^{\int 2dx}=e^{2x}$.
2. Multiply ODE: $e^{2x}\frac{dy}{dx}+2e^{2x}y=e^x$.
3. Left side: $\frac{d}{dx}(e^{2x}y)=e^x$.
4. Integrate: $e^{2x}y=\int e^{x}dx=e^x+C$.
5. Solve: $y=e^{-x}+C{e}^{-2x}$.
6. Apply IC: $0=1+C\Rightarrow C=-1$. Final Answer: $y=e^{-x}-e^{-2x}$.

Verification: Substitute $y$ back into the ODE: $$\frac{dy}{dx}=-e^{-x}+2e^{-2x},2y=2e^{-x}-2e^{-2x}.$$ LHS: $-e^{-x}+2e^{-2x}+2e^{-x}-2e^{-2x}=e^{-x}=\text{RHS}$.

---

3.3 Exact Equations

Definition: 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 Method:

1. Check exactness.
2. Find $\psi (x,y)$ such that: $$\frac{\partial \psi }{\partial x}=M\text{and}\frac{\partial \psi }{\partial y}=N.$$
3. Integrate $M$ w.r.t. $x$, then adjust for $N$.
4. Solve $\psi (x,y)=C$.

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

Solution:

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

Verification: Differentiate implicitly: $$(2xy+3)dx+(x^2+4y)dy=0\text{(matches the original)}.$$

---

3.4 Homogeneous Equations

Definition: $\frac{dy}{dx}=f\left(\frac{y}{x}\right)$. Solve using substitution $v=\frac{y}{x}$.

Steps:

1. Let $y=vx$, then $\frac{dy}{dx}=v+x\frac{dv}{dx}$.
2. Substitute into ODE: $v+x\frac{dv}{dx}=f(v)$.
3. Separate variables and solve for $v$.
4. Back-substitute $v=\frac{y}{x}$.

Example 4: Solve $\frac{dy}{dx}=\frac{x^2+y^2}{xy}$.

Solution:

1. Substitute $y=vx$: $$v+x\frac{dv}{dx}=\frac{x^2+v^2{x}^2}{x\cdot vx}=\frac{1+v^2}{v}.$$
2. Separate: $$x\frac{dv}{dx}=\frac{1+v^2}{v}-v=\frac{1}{v}.$$
3. Integrate: $$\int vdv=\int \frac{1}{x}dx\Rightarrow \frac{v^2}{2}=\ln |x|+C.$$
4. Back-substitute $v=\frac{y}{x}$: $$\frac{y^2}{2x^2}=\ln |x|+C\Rightarrow y^2=2x^2\ln |x|+C{x}^2.$$

Verification: Differentiate implicitly and substitute back into the ODE to confirm.

---

3.5 Bernoulli Equations

Form: $\frac{dy}{dx}+P(x)y=Q(x)y^n$. Solution: Substitute $v=y^{1-n}$, reducing to linear.

Example 5: Solve $\frac{dy}{dx}+y=y^2\sin x$.

Solution:

1. Here, $n=2$, so $v=y^{-1}$, $\frac{dv}{dx}=-y^{-2}\frac{dy}{dx}$.
2. Rewrite ODE: $$\frac{dy}{dx}=y^2\sin x-y\Rightarrow -y^{-2}\frac{dy}{dx}=\sin x-y^{-1}.$$
3. Substitute $v$: $$\frac{dv}{dx}=\sin x-v.$$
4. Linear ODE: $\frac{dv}{dx}+v=\sin x$.
5. Solve using integrating factor:
   • $\mu (x)=e^{\int 1dx}=e^x$.
   • Multiply: $e^x\frac{dv}{dx}+e^{x}v=e^x\sin x$.
   • Integrate: $e^{x}v=\int e^x\sin xdx$.
6. Use integration by parts for $\int e^x\sin xdx$: $$\int e^x\sin xdx=\frac{e^x(\sin x-\cos x)}{2}+C.$$
7. Thus: $$v=\frac{\sin x-\cos x}{2}+C{e}^{-x}.$$
8. Back-substitute $v=\frac{1}{y}$: $$y=\frac{1}{\frac{\sin x-\cos x}{2}+C{e}^{-x}}.$$

---

4. Applications of First-Order ODEs

First-order ODEs model real-world phenomena where rates depend on current state.

4.1 Population Growth (Logistic Model)

Model: $$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),$$ where:

• $P(t)$: Population at time $t$,
• $r$: Growth rate,
• $K$: Carrying capacity.

Solution: Separable equation. Solution is the logistic function: $$P(t)=\frac{K}{1+\left(\frac{K-P_0}{P_0}\right)e^{-rt}}.$$

4.2 Newton’s Law of Cooling

Model: $$\frac{dT}{dt}=-k(T-T_{\text{env}}),$$ where:

• $T(t)$: Temperature of object at time $t$,
• $T_{\text{env}}$: Ambient temperature,
• $k$: Cooling constant.

Solution: Linear ODE. Solution: $$T(t)=T_{\text{env}}+(T_0-T_{\text{env}})e^{-kt}.$$

4.3 Orthogonal Trajectories

Definition: Curves that intersect a given family of curves at right angles. Method:

1. Solve the family of curves $F(x,y,C)=0$ for $\frac{dy}{dx}$.
2. Orthogonal trajectories satisfy $\frac{dy}{dx}=-\frac{dx}{dy}$.
3. Solve the new ODE.

Example 6: Find orthogonal trajectories to $x^2+y^2=C$.

Solution:

1. Differentiate implicitly: $2x+2y\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=-\frac{x}{y}$.
2. Orthogonal slope: $\frac{dy}{dx}=\frac{y}{x}$.
3. Solve separable ODE: $$\frac{dy}{y}=\frac{dx}{x}\Rightarrow \ln |y|=\ln |x|+C\Rightarrow y=Cx.$$ Answer: Orthogonal trajectories are straight lines through the origin.

---

5. Comparison of Methods

---

6. Common Mistakes and Pitfalls

1. Assuming All ODEs Are Separable: Always check the form before applying separation.
2. Forgetting the Integrating Factor: For linear ODEs, $\mu (x)$ is essential.
3. Exactness Check: Skipping $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ leads to incorrect solutions.
4. Initial Conditions: Always apply ICs to find particular solutions.
5. Substitution Errors: In homogeneous/Bernoulli equations, ensure correct substitution and back-substitution.

---

7. Exam Tips

7.1 What Examiners Look For

• Correct Classification: Identify the type of ODE (separable, linear, etc.) before solving.
• Step-by-Step Solutions: Show all steps, especially for integrating factors and substitutions.
• Verification: Always verify your solution by substituting back into the original ODE.
• Initial Conditions: If given, apply them correctly to find $C$.
• Simplification: Simplify final answers (e.g., $y=e^x+C$ instead of $y=C{e}^x+e^x$).

7.2 Common Exam Questions

1. Solve the following ODEs:
   • Separable: $\frac{dy}{dx}=x{y}^2$.
   • Linear: $\frac{dy}{dx}+3y=\sin x$.
   • Exact: $(3x^2+2y)dx+(2x-4y)dy=0$.
2. Applications:
   • Model population growth with $\frac{dP}{dt}=0.1P$.
   • Solve Newton’s cooling problem with given $T(0)$ and $T_{\text{env}}$.
3. Orthogonal Trajectories: Find trajectories orthogonal to $y=e^x$.

7.3 High-Scoring Strategies

• Show Work: Even if the answer is simple, write steps (e.g., integrating factor calculation).
• Box Final Answers: Clearly mark the final solution (e.g., $y=\boxed{e^{x^2}}$).
• Units and ICs: If an application problem is given, include units and interpret constants.
• Alternative Methods: If stuck, try rewriting the ODE (e.g., exact vs. integrating factor).

7.4 Past Exam Patterns (Based on Given Questions)

• Misleading Questions: Some questions (e.g., $\frac{\partial z}{\partial x}$) are not first-order ODEs but partial derivatives. Ignore them for this unit.
• Numerical Methods: Questions like "Find $x_3$ using Newton-Raphson" belong to Unit 8 (Numerical Methods), not this unit.
• Focus on Solving: Exams prioritize solving separable, linear, and exact ODEs. Practice these heavily.

---

8. Practice Problems

Solve the following (answers provided at the end):

1. Separable: $\frac{dy}{dx}=e^{x+y}$.

2. Linear: $\frac{dy}{dx}+\frac{y}{x}=x^2$.

3. Exact: $(y+2x)dx+(x-y)dy=0$.

4. Homogeneous: $\frac{dy}{dx}=\frac{x+y}{x-y}$.

5. Application: A tank contains 1000 L of water with 50 kg of salt. Water with 1 kg/L of salt enters at 5 L/min, and mixed water leaves at 5 L/min. Find $S(t)$, the amount of salt at time $t$.

---

Answers to Practice Problems

1. $y=-e^{-x}+C$.
2. $y=\frac{x^3}{4}+C{x}^2$.
3. $x^2-y^2+2xy=C$.
4. $x^2+y^2=Cx$.
5. $S(t)=50+5000t-5000e^{-t/200}$ kg.

---

9. Summary Table of Solution Methods

---

10. Final Notes

• Memorize Forms: Know the standard forms of separable, linear, exact, and homogeneous ODEs.
• Practice Verification: Always plug your solution back into the original ODE.
• Time Management: In exams, start with separable/linear ODEs (easier) before attempting exact/homogeneous.
• Applications: Understand the physical meaning behind models (e.g., cooling, growth).

---

Exam Tip

"If the ODE doesn’t fit separable, linear, or exact, try homogeneous or Bernoulli. If all else fails, check for exactness after multiplying by an integrating factor."

---

End of Note

This covers all subtopics of Unit 6 as per the TU syllabus, with definitions, methods, examples, applications, and exam strategies. For further practice, refer to past TU/PU question papers focusing on first-order ODEs.