Question: Convert the Differential Equations lecture into a step-by-step interactive blackboard teaching session.Do not generate textbook notes or long paragraphs.I want you to TEACH by solving mathematics on the screen.Start with:1. Order of differential equations.2. Degree of differential equations.3. Formation of differential equations.4. Variable separable method.For every topic:- Write the mathematical problem on screen.- Solve it line by line.- Show every differentiation and integration step.- Explain each step through narration.- Highlight the formulas being used.- Give 3 solved examples of increasing difficulty.- Give practice questions.Use a digital blackboard with handwritten-style mathematical equations.STRICTLY AVOID:- Unrelated images.- Stock footage.- Decorative animations.- Textbook-style paragraphs.- Skipping mathematical calculations.The screen must primarily show mathematical equations and their solutions.
1. **Order of Differential Equations**
The order of a differential equation is the order of the highest derivative present in the equation.
**Example:** Find the order of $$\frac{d^3y}{dx^3} + 5\frac{dy}{dx} + y = 0$$
- The highest derivative is $$\frac{d^3y}{dx^3}$$ which is the third derivative.
- So, the order is 3.
**Solved Examples:**
1. $$\frac{d^2y}{dx^2} + 3y = 0$$ Order is 2.
2. $$\frac{dy}{dx} + y = e^x$$ Order is 1.
3. $$\frac{d^4y}{dx^4} - x\frac{d^2y}{dx^2} + y = 0$$ Order is 4.
**Practice Questions:**
1. Find the order of $$\frac{d^5y}{dx^5} + y = 0$$
2. Find the order of $$\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0$$
2. **Degree of Differential Equations**
The degree of a differential equation is the power of the highest order derivative after the equation is free from radicals and fractions with derivatives.
**Example:** Find the degree of $$\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} = 0$$
- Highest order derivative is $$\frac{d^2y}{dx^2}$$
- Its power is 3.
- So, degree is 3.
**Solved Examples:**
1. $$\left(\frac{dy}{dx}\right)^2 + y = 0$$ Degree is 2.
2. $$\sqrt{\frac{d^3y}{dx^3}} + y = 0$$ Not a polynomial in derivatives, degree not defined.
3. $$\left(\frac{d^4y}{dx^4}\right)^5 + x = 0$$ Degree is 5.
**Practice Questions:**
1. Find the degree of $$\left(\frac{d^2y}{dx^2}\right)^4 + y = 0$$
2. Find the degree of $$\frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^{\frac{1}{2}} = 0$$
3. **Formation of Differential Equations**
Form a differential equation from the family of curves $$y = c_1 e^{2x} + c_2 e^{-3x}$$
- Differentiate once:
$$\frac{dy}{dx} = 2 c_1 e^{2x} - 3 c_2 e^{-3x}$$
- Differentiate twice:
$$\frac{d^2y}{dx^2} = 4 c_1 e^{2x} + 9 c_2 e^{-3x}$$
- Multiply first derivative by 3 and add to second derivative:
$$\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} = 4 c_1 e^{2x} + 9 c_2 e^{-3x} + 6 c_1 e^{2x} - 9 c_2 e^{-3x} = 10 c_1 e^{2x}$$
- Multiply original equation by -5:
$$-5 y = -5 c_1 e^{2x} - 5 c_2 e^{-3x}$$
- Combine:
$$\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} - 5 y = 0$$
**Solved Examples:**
1. From $$y = c_1 e^x + c_2 e^{-x}$$ form the differential equation.
2. From $$y = c_1 \sin x + c_2 \cos x$$ form the differential equation.
3. From $$y = c_1 x + c_2$$ form the differential equation.
**Practice Questions:**
1. Form the differential equation from $$y = c_1 e^{3x} + c_2 e^{-x}$$
2. Form the differential equation from $$y = c_1 \cos 2x + c_2 \sin 2x$$
4. **Variable Separable Method**
Solve $$\frac{dy}{dx} = y \sin x$$
- Separate variables:
$$\frac{1}{y} dy = \sin x dx$$
- Integrate both sides:
$$\int \frac{1}{y} dy = \int \sin x dx$$
- Integrate:
$$\ln |y| = -\cos x + C$$
- Exponentiate both sides:
$$|y| = e^{-\cos x + C} = e^C e^{-\cos x}$$
- Let $$A = e^C$$, so:
$$y = A e^{-\cos x}$$
**Solved Examples:**
1. Solve $$\frac{dy}{dx} = xy$$
2. Solve $$\frac{dy}{dx} = \frac{y}{x}$$
3. Solve $$\frac{dy}{dx} = (1 + y^2) x$$
**Practice Questions:**
1. Solve $$\frac{dy}{dx} = y \cos x$$
2. Solve $$\frac{dy}{dx} = \frac{x}{y}$$