Differential Equations Easy Written Notes – Free Download PDF!

Struggling with differential equations (DE)? These free PDF notes simplify DEs with easy explanations, solved examples, and real-world applications—perfect for engineering, physics, and math students!

What’s Inside the PDF?

1. Basics of Differential Equations

• What is a DE? An equation with derivatives (e.g., $\frac{dy}{dx}=x^2$).

• Types:

  • Ordinary DE (ODE): 1 independent variable (e.g., $y^{\mathrm{\prime }\mathrm{\prime }}+y=0$).

  • Partial DE (PDE): Multiple variables (e.g., $\frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}=0$).

• Order & Degree:

  • Order: Highest derivative (e.g., $y^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}$ → 3rd order).

  • Degree: Power of the highest derivative (e.g., $(y^{\mathrm{\prime }\mathrm{\prime }}{)}^2$ → 2nd degree).

2. Solving First-Order DEs

A. Separation of Variables

• Split $x$ and $y$ terms, then integrate:

  $$\frac{dy}{dx}=xy\Rightarrow \int \frac{dy}{y}=\int xdx$$

  Example: Solve $\frac{dy}{dx}=e^{x}y$ → $y=C{e}^{e^x}$.

B. Homogeneous DEs

• Use substitution $y=vx$:

  $$\frac{dy}{dx}=\frac{x^2+y^2}{xy}\Rightarrow \text{Let }y=vx\text{ and simplify.}$$

C. Exact DEs

• Check if $\frac{\mathrm{\partial }M}{\mathrm{\partial }y}=\frac{\mathrm{\partial }N}{\mathrm{\partial }x}$, then solve:

  $$(3x^2+y)dx+(x+2y)dy=0\text{(Exact!)}$$

D. Linear DEs

• Use integrating factor $e^{\int P(x)dx}$:

  $$\frac{dy}{dx}+2y=4x\Rightarrow y=2x-1+C{e}^{-2x}\mathrm{.}$$

3. Higher-Order DEs

A. Linear with Constant Coefficients

• Solve $y^{\mathrm{\prime }\mathrm{\prime }}-5y^{\mathrm{\prime }}+6y=0$:

  1. Find roots: $m^2-5m+6=0$ → $m=2,3$.

  2. General solution: $y=C_1{e}^{2x}+C_2{e}^{3x}$.

B. Non-Homogeneous DEs

• Add particular solution ($y_p$) to general solution ($y_h$):

  $$y^{\mathrm{\prime }\mathrm{\prime }}+y=\sin x\Rightarrow y_p=-\frac{x}{2}\cos x\mathrm{.}$$

Real-World Applications

1. Physics (Motion)

• Newton’s 2nd Law: $F=ma=m\frac{d^{2}x}{d{t}^2}$.

• Projectile Motion: Solve DEs for velocity/position.

2. Radioactive Decay

• DE Model: $\frac{dQ}{dt}=-kQ$ → $Q=Q_0{e}^{-kt}$.

• Half-life: Time for substance to decay to 50%.

3. Population Growth

• DE Model: $\frac{dP}{dt}=kP$ → $P=P_0{e}^{kt}$.

4. Electrical Circuits (RLC)

• DE Model: $L\frac{d^{2}I}{d{t}^2}+R\frac{dI}{dt}+\frac{I}{C}=E(t)$.

Who Should Download This?

🎓 Engineering Students (Mechanical, Electrical, Civil)
🔬 Physics & Math Majors
📚 Teachers & Competitive Exam Aspirants (GATE, GRE, FE)

Why These Notes?

✔ Step-by-Step Solutions – No skipped steps!
✔ Diagrams & Examples – Visual learning.
✔ Free PDF Download – No signup needed.

Final Thoughts

Differential equations are everywhere—from rocket science to bank loans. Mastering them opens doors to advanced engineering and science!

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