text: # Thread: Use integral calculus to solve differential equation problems
  
  1. ## Use integral calculus to solve differential equation problems
  
  Find the equation of the curve that satisfies the differential equation f"(x) = 6x - 12 and which passes through the points (0, -8) and (1, -1).
  
  2. Originally Posted by LilDragonfly
  
  Find the equation of the curve that satisfies the differential equation f"(x) = 6x - 12 and which passes through the points (0, -8) and (1, -1).
  $f(x)=x^3-6x^2+ax+b$
  Sooooo:
  
  $f(0)=-8 = 0^3-6*0^2+a*0+b=b$
  Thus b = -8.
  
  $f(1)=-1=(1)^3-6(1)^2+a(1)-8$
  gives:
  $-1=1-6+a-8=-13-a$
  or
  $a = 12$
  
  -Dan
  
  3. I do not know if you understood what topsquak said, but when you have,
  $f''(x)=6x-12$
  It means if you integrate both sides you have,
  $f'(x)=3x^2-12x+a$ with some constant $a$,
  the reason to why f''(x) becomes f'(x) is because when you integrate a derivative of a function you get back the function (definition of integral). So when you have f''(x) integrating once would yield f'(x) so you need one more time:
  $f(x)=x^3-6x^2+ax+b$
  Then use topsqaurk post to complete the problem.
metadata: {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}