Differential Equations

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A differential equation is an equation that involves one or more derivatives of a dependent variable with respect to an independent variable. Definition: An equation containing derivatives such as dy

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Step 1: Remove fractional powers by squaring both sides [1 + (dy/dx)²]³ = (d²y/dx²)² Step 2: Identify the highest order derivative Highest derivative: d²y/dx² (second order) Step 3: Find the power o

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ORDER: The order of the highest derivative occurring in the equation. DEGREE: The degree (power) of the highest order derivative after the equation is free from radicals and fractions. Example: (d²y

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Step 1: Check if dependent variable and its derivatives appear in first power only The term y(d²y/dx²) contains the product of y and d²y/dx² Step 2: Check if derivatives are multiplied together Here

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Step 1: Count arbitrary constants Constants: a and b (2 constants) Expected order: 2 Step 2: Differentiate once dy/dx = 2ax + b ... (1) Step 3: Differentiate again d²y/dx² = 2a ... (2) Step 4: Elim

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GENERAL SOLUTION: • Contains as many arbitrary constants as the order of the differential equation • Represents the complete family of curves • Example: y = c₁e^x + c₂e^(-x) for d²y/dx² - y = 0 PARTI

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Step 1: Identify the type This is of the form dy/dx = f(x)g(y) (variables separable) Here: e^(x-y) = e^x · e^(-y) Step 2: Separate variables dy/dx = e^x · e^(-y) e^y dy = e^x dx Step 3: Integrate bo

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Step 1: Separate variables (1 + x²)dy = (1 + y²)dx dy/(1 + y²) = dx/(1 + x²) Step 2: Integrate both sides ∫dy/(1 + y²) = ∫dx/(1 + x²) Step 3: Apply standard integrals ∫1/(1 + y²)dy = tan⁻¹y + c₁ ∫1/