Linear Programming

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A differential equation is an equation that contains derivatives of a function. Examples: 1. dy/dx = cos x (first order) 2. d²y/dx² + k²y = 0 (second order) 3. d³y/dx³ + x(dy/dx) - 4xy = 0 (third ord…

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ORDER: The highest order of derivative in the equation DEGREE: The power of the highest ordered derivative Steps: 1. Identify the highest order derivative 2. Check if equation is polynomial in deriva…

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Step 1: Eliminate the square root by squaring both sides 1 + (dy/dx)² = (d²y/dx²)² Step 2: Rearrange (dy/dx)² + 1 = (d²y/dx²)² Step 3: Identify order and degree - Highest order derivative: d²y/dx² -…

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Step 1: Given equation y = ax² Step 2: Differentiate with respect to x dy/dx = 2ax Step 3: From original equation, a = y/x² Step 4: Substitute in the derivative dy/dx = 2(y/x²)x = 2y/x Step 5: Rea…

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Step 1: Given y = Ae³ˣ + Be⁻³ˣ (two constants A, B) Step 2: First differentiation dy/dx = 3Ae³ˣ - 3Be⁻³ˣ Step 3: Second differentiation d²y/dx² = 9Ae³ˣ + 9Be⁻³ˣ = 9(Ae³ˣ + Be⁻³ˣ) Step 4: From origi…

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METHOD: Separate variables x and y on different sides of the equation Condition: Equation must be writable as f(x)dx = g(y)dy Steps: 1. Rearrange to separate variables 2. Integrate both sides 3. Add…

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Step 1: Separate variables dy = x√(25 - x²) dx Step 2: Integrate both sides ∫dy = ∫x√(25 - x²) dx Step 3: For RHS, use substitution Let 25 - x² = t -2x dx = dt x dx = -dt/2 Step 4: Substitute ∫dy =…

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DEFINITION: A differential equation where all terms have the same degree Identification: 1. Replace dy/dx with a constant k 2. Check if all terms have same degree in x and y Examples: ✓ Homogeneous:…