Differential Equation and Applications

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A differential equation is an equation involving dependent variable(s), independent variable(s), and derivatives of the dependent variable(s) with respect to the independent variable. Examples: 1) dy…

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The ORDER is the highest order derivative present in the equation. Examples: • dy/dx = y + x → Order = 1 (highest derivative is first order) • d²y/dx² + 3(dy/dx) - 2y = 0 → Order = 2 (highest derivat…

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Step-by-step method: 1) Make all derivatives free from fractional/negative indices 2) Identify the highest order derivative 3) Find its power For (d²y/dx²)³ = 2 + (dy/dx)²: • Already free from fracti…

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GENERAL SOLUTION: • Contains arbitrary constants • Number of constants = order of equation • Example: y = Ae^x + Be^(-x) (2 constants for 2nd order) PARTICULAR SOLUTION: • Obtained by giving specific…

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Step 1: Separate variables dy/(1+y) = dx/(1+x) Step 2: Integrate both sides ∫dy/(1+y) = ∫dx/(1+x) Step 3: Apply integration log(1+y) = log(1+x) + log c log(1+y) = log[c(1+x)] Step 4: Remove logarit…

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Use substitution when variables cannot be separated directly. Step 1: Rearrange dy/dx = (2x - 2y + 3)/(x - y + 1) = (2(x-y) + 3)/(x-y+1) Step 2: Substitute x - y = t dt/dx = 1 - dy/dx, so dy/dx = 1 …

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DEFINITION: A differential equation f(x,y)dx + g(x,y)dy = 0 is homogeneous if f(x,y) and g(x,y) are homogeneous functions of the same degree. IDENTIFICATION TEST: Replace x with λx and y with λy in b…

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Step 1: Verify it's homogeneous Both x² + y² and 2xy have degree 2 ✓ Step 2: Rearrange dy/dx = (x² + y²)/(2xy) Step 3: Substitute y = tx, dy/dx = t + x(dt/dx) t + x(dt/dx) = (x² + t²x²)/(2x·tx) = (1…