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Solving Differential Equation using Slope Fields and Integration

Learn how to solve a differential equation step by step using slope fields, integration, and variable separation method. Understand the process with an example starting from a random point. Find the value of C and explore exponential equations.

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Solving Differential Equation using Slope Fields and Integration

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  1. dy/dx = dy/dx = y-1 2-1 x2 22 (1/4) (-1/4) Part (a) Let’s start with a random point—(2,2) for example... Repeating this process8 more times gives usour slope field. m=1/4

  2. dy/dx = y-1 x2 ln y-1 = -x-1 + C ln y-1 = -x-1 + 1/2 ln 0-1 = -2-1 + C -1 x ( +½) ln y-1 e = e Part (b) Separate the variables (y-1)-1= x-2 dx Integrate both sides Find the value of C 0 = -1/2 + C C = 1/2 Exponentiate both sides

  3. -1 ( +½) 2  e = 0 1 = e y-1  e0 = 0 1  e = y-1  e = y 1 e = 1- y -1 -1 -1 -1 -1 x x x x x ( +½) ( +½) ( +½) ( +½) ( +½) ln y-1 Part (b) 1 Use the negative part. e = e

  4. lim e 1- x 1 – e = e = 1- y -1 -1 x x ( +½) ( +½) Part (c) = 1 – e0+½ = 1 – e½

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