A formula that fives prime numbers



Case 1: dydx=fxSolve the differential equation dydx=4x3-6x2+11, given that y=8 when x=1.Method 1: Indefinite Integral techniqueWorked solutionExplanationdy=4x3-6x2+11.dxMultiply both sides of the equation by dx so that the variables are separated.dy=4x3-6x2+11.dx Integrate both sides of the equation.y=4x44-6x33+11x+CAcknowledge the constant on the right-hand side of the equation only.y=x4-2x3+11x+C ?Simplify the expression.8=14-213+111+CSubstitute in y=8 when x=1.C=8-14+213-11(1)C=-2 Rearrange to find C.∴y=x4-2x3+11x-2 Substitute back into ?Method 2: Definite integral techniqueWorked solutionExplanationdy=4x3-6x2+11.dxMultiply both sides of the equation by dx so that the variables are separated.y=8ydy=x=1x4x3-6x2+11.dxIntegrate both sides of the equation by forming an integral with variables x and y as the upper limits and the conditions y=8 and x=1 as the lower limits. y8y=4x44-6x33+11x1xThere is no need to consider constant values using this method.y-8=x4-2x3+11x-14-213+111Evaluate both integral statements using the limits.∴y=x4-2x3+11x-2 Rearrange to find the solution. ................
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