JEE Main Level

1. Formation of Differential Equation

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

1. Eliminate constants by introducing new variables $x=at+b, \ y=cy+d$.
2. Differentiate w.r.t new variable $x$, $y,$ and eliminate the parameter.
3. 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)