### Shortcut Methods

**JEE Main Level**

**1. Formation of Differential Equation**

**Shortcut:** To form ODEs from functions, follow the steps:

- Eliminate constants by introducing new variables $x=at+b, \ y=cy+d$.
- Differentiate w.r.t new variable $x$, $y,$ and eliminate the parameter.
- Simplify using properties of derivatives.

**2. Order and Degree of Differential Equation**

**Shortcut:**

- Count the order of highest order derivative present.
- Count the degree of highest power in which highest order derivative is raised.

**3. Solution of Differential Equations**

**Shortcut:**

- Use separation of variables for first-order equations.
- Check for exact differential equations.
- Use integrating factors.
- Use substitution methods.

**4. Applications of Differential Equations**

**Shortcut:**

- Use ODEs to model physical phenomena like projectile motion.
- Use ODEs to solve problems with population dynamics, radioactive decay, and more.
- Use partial differential equations to model heat/wave equations, fluid dynamics, elastic materials.

**5. Linear Differential Equation**

**Shortcut:**

- For linear equations of the form (ay’’+by+c=0), identify (m_{1,2}) and (y_c) then combine in (y_c+y_p), where (y_p) is a particular solution.

**6. Homogenous Differential Equations**

**Shortcut:**

- For homogenous equations of the form (ay’’+by+c=0), substitute (y=A{e^{mx}}) and solve for roots (m).

**7. Exact Differential Equations**

**Shortcut:**

- Check for (M_x=N_y). If true, find the potential function (P) such that (M=\frac{\delta P}{\delta x}, \ N=\frac{\delta P}{\delta y}).

**8. Variable Separable Differential Equations**

**Shortcut:**

- For variable separable equations of the form (M(y)dy=N(x)dx), rearrange to separate variables and integrate.

**Numerical Examples:**

1. (y=Ce^{\tan^{-1}x})

2. (ye^{-x}+4x^3=C)

3. (y=e^x(\cos2x+\sin2x+1))

**CBSE Board Exam Level**

**1. Formation of Differential Equation**

**Shortcut:**

- Eliminate constants by introducing new variables.
- Differentiate w.r.t the new variable.
- Simplify using derivative properties.

**2. Solution of Differential Equations**

**Shortcut:**

- Apply direct integration
- Apply the method of integrating factor
- Use formulas for first and second-order equations.

**3. Applications of Differential Equations**

**Shortcut:**

- Identify the relevant concept (projectile motion, population growth, etc.).
- Set up the differential equation, solve, and interpret the results.

**4. Linear Differential Equations of First and Second Order**

**Shortcut:**

- For linear equations of the first order of the form (ay’+by=c), use the integrating factor (e^{\int \frac{b}{a}dx}).
- For linear equations of the second order of the form (ay’’+by’+c=0), use the auxiliary equation.

**Numerical Examples:**

1. (y=x^2 (x-1)^2 + C)

2. (y=Ce^x+x-1)

3.(y=c_1\cos x+c_2\sin x+1)