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Solving Initial Value Problems

Solving Initial Value Problems. Vanessa Hernandez Jordan Hughes Period 3. Things You Will Learn. Definition of Initial Value Problem Antiderivative rules How to solve for C How to completely answer initial value problems. Definition.

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Solving Initial Value Problems

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  1. Solving Initial Value Problems Vanessa Hernandez Jordan Hughes Period 3

  2. Things You Will Learn • Definition of Initial Value Problem • Antiderivative rules • How to solve for C • How to completely answer initial value problems

  3. Definition • An initial value problem is a differential equation that needs a specific answer. • When solving for the initial condition you are given a the derivative and a point on the original function.

  4. Example #1 • y’= (x-1)³, y(0)=2 • y=¼(x-1)⁴+C • 2=¼(0-1) ⁴ • Find the Antiderivative by: • First add 1 to the power and divide by 4 making the equation y=¼(x-1)⁴+C. • Plug in the value for y(0)=2, giving you 2=¼(0-1) ⁴.

  5. Example #1 Cont. • 2=¼(-1) ⁴+C • 2=¼+C 3) Continue to simplify the equation 4)Subtract ¼ to the other side of the equation to solve for C.

  6. Example #1 Cont. • C=7/4 or 1.75 • y=¼(x-1) ⁴+1.75 5) The value for C is found to be 7/4 or 1.75. 6) The answer to the complete equation is y=¼(x-1) ⁴+1.75.

  7. Example #2 Find the antiderivative by: Raise the power from 5 to 6, then divide 2 by 6. Which gives you ⅓x⁶+ C. Plug in the value given, y(6)=3, to obtain the value of C. y’= 2x⁵ , y(6)=3 y=⅓x⁶+ C 3= ⅓(6)⁶+ C

  8. Example #2 Cont. • 3 = 15552+C • -15549 = C • y = ⅓x ⁶-15549 3) Calculate the value, then subtract 15552 to solve for C. 4) The C value is found to be -15549. 5) The equation is y= ⅓x ⁶-15549

  9. Example #3 Find the antiderivative by: Raising the equation by one power and then dividing by that power. It is also necessary to add a constant (C). Then you must plug in the initial condition given. * Notice the x was substituted for 0 and the whole equation was set equal to 7. • y’=2sinx; y(0)=7 • y= sin²x + C • y= sin((0))² + C=7

  10. Example #3 Cont’d • y’=2sinx; y(0)=7 • y= (sin(0))² + C=7 • (0)+ C=7 • C=7 • y= sin²x+7 3) Now solve for C. The constant in the antiderivative equals 7. 4) The answer to the equation is y= sin²x+7

  11. Important Reminders • Don’t forget to solve for and include C in your answer. • If you don’t solve for C, the problem is incomplete. • Realize that you are taking the antiderivative, NOT the derivative. • Don’t forget to divide by the raised power.

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