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المعادلات التفاضلية

سائل كيميائيتتغير درجة حرارة سائل كيميائي بارد، بعد وضعه في غرفة دافئة، بمعدل يمكن نمذجته بالمعادلة التفاضلية: begin mathsize 20px style fraction numerator d A over denominator d t end fraction equals 2 left parenthesis 20 minus A right parenthesis end style، حيث begin mathsize 20px style A end style درجة حرارة السائل بمقياس سيلسيوس، وbegin mathsize 20px style t end style الزمن بالساعات:

(1) أحل المعادلة التفاضلية لإيجاد درجة حرارة السائل بعد begin mathsize 20px style t end style ساعة، علماً بأن درجة حرارته عند وضعه في الغرفة هي begin mathsize 20px style 5 to the power of ring operator straight C end style.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 1 over denominator x cubed plus x end fraction equals fraction numerator 1 over denominator x left parenthesis x squared plus 1 right parenthesis end fraction equals A over x plus fraction numerator B x plus C over denominator x squared plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow 1 equals A left parenthesis x squared plus 1 right parenthesis plus left parenthesis B x plus C right parenthesis left parenthesis x right parenthesis end cell row blank cell x equals 0 not stretchy rightwards double arrow A equals 1 end cell row blank cell x equals 1 not stretchy rightwards double arrow 1 equals 2 A plus B plus C not stretchy rightwards double arrow 1 equals 2 plus B plus C end cell row blank cell x equals negative 1 not stretchy rightwards double arrow 1 equals 2 A plus B minus C not stretchy rightwards double arrow 1 equals 2 plus B minus C end cell end table end style

إذن، يمكن نمذدجة درجة الحرارة C بعد t ساعة بالعلاقة:

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell A equals integral subscript 1 superscript 2 fraction numerator 1 over denominator x cubed plus x end fraction d x end cell cell equals integral subscript 1 superscript 2 left parenthesis 1 over x plus fraction numerator negative x over denominator x squared plus 1 end fraction right parenthesis d x end cell row blank cell equals l n invisible function application vertical line x vertical line minus 1 half l n invisible function application vertical line x squared plus 1 vertical line vertical line subscript 1 superscript 2 end cell row blank cell equals l n invisible function application 2 minus 1 half l n invisible function application 5 minus l n invisible function application 1 plus 1 half l n invisible function application 2 end cell row blank cell equals 3 over 2 l n invisible function application 2 minus 1 half l n invisible function application 5 equals 1 half l n invisible function application 8 over 5 end cell end table end style

(2) بعد كم ساعة تصبح درجة حرارة السائل begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell A equals integral subscript 1 superscript 2 fraction numerator 1 over denominator x cubed plus x end fraction d x end cell cell equals integral subscript 1 superscript 2 left parenthesis 1 over x plus fraction numerator negative x over denominator x squared plus 1 end fraction right parenthesis d x end cell row blank cell equals l n invisible function application vertical line x vertical line minus 1 half l n invisible function application vertical line x squared plus 1 vertical line vertical line subscript 1 superscript 2 end cell row blank cell equals l n invisible function application 2 minus 1 half l n invisible function application 5 minus l n invisible function application 1 plus 1 half l n invisible function application 2 end cell row blank cell equals 3 over 2 l n invisible function application 2 minus 1 half l n invisible function application 5 equals 1 half l n invisible function application 8 over 5 end cell end table end style؟ 

نعوض begin mathsize 20px style A equals 18 end style في العلاقة begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell A equals integral subscript 1 superscript 2 fraction numerator 1 over denominator x cubed plus x end fraction d x end cell cell equals integral subscript 1 superscript 2 left parenthesis 1 over x plus fraction numerator negative x over denominator x squared plus 1 end fraction right parenthesis d x end cell row blank cell equals l n invisible function application vertical line x vertical line minus 1 half l n invisible function application vertical line x squared plus 1 vertical line vertical line subscript 1 superscript 2 end cell row blank cell equals l n invisible function application 2 minus 1 half l n invisible function application 5 minus l n invisible function application 1 plus 1 half l n invisible function application 2 end cell row blank cell equals 3 over 2 l n invisible function application 2 minus 1 half l n invisible function application 5 equals 1 half l n invisible function application 8 over 5 end cell end table end style فينتج:

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell A equals integral subscript 1 superscript 2 fraction numerator 1 over denominator x cubed plus x end fraction d x end cell cell equals integral subscript 1 superscript 2 left parenthesis 1 over x plus fraction numerator negative x over denominator x squared plus 1 end fraction right parenthesis d x end cell row blank cell equals l n invisible function application vertical line x vertical line minus 1 half l n invisible function application vertical line x squared plus 1 vertical line vertical line subscript 1 superscript 2 end cell row blank cell equals l n invisible function application 2 minus 1 half l n invisible function application 5 minus l n invisible function application 1 plus 1 half l n invisible function application 2 end cell row blank cell equals 3 over 2 l n invisible function application 2 minus 1 half l n invisible function application 5 equals 1 half l n invisible function application 8 over 5 end cell end table end style

إذن، تصبح درجة حرارة السائل begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell A equals integral subscript 1 superscript 2 fraction numerator 1 over denominator x cubed plus x end fraction d x end cell cell equals integral subscript 1 superscript 2 left parenthesis 1 over x plus fraction numerator negative x over denominator x squared plus 1 end fraction right parenthesis d x end cell row blank cell equals l n invisible function application vertical line x vertical line minus 1 half l n invisible function application vertical line x squared plus 1 vertical line vertical line subscript 1 superscript 2 end cell row blank cell equals l n invisible function application 2 minus 1 half l n invisible function application 5 minus l n invisible function application 1 plus 1 half l n invisible function application 2 end cell row blank cell equals 3 over 2 l n invisible function application 2 minus 1 half l n invisible function application 5 equals 1 half l n invisible function application 8 over 5 end cell end table end style بعد مرور ساعة واحدة تقريباً بعد وضعه في الغرفة.

إعداد : شبكة منهاجي التعليمية

12 / 02 / 2023

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