Inne

Sposób rozwiązywania tego typu równań (drugiego rzędu sprowadzalne do pierwszego przez podstawienie) przedstawiłem w tej Lekcji.

Pójdzie tak:

$y double apostrophe equals 1 minus open parentheses y apostrophe close parentheses squared open parentheses y apostrophe close parentheses apostrophe equals 1 minus open parentheses y apostrophe close parentheses squared$

Stosuję teraz podstawienie: $t equals y apostrophe$

$t apostrophe equals 1 minus t squared$

I dalej już ciurkiem metodą zmiennych rozdzielonych:

$fraction numerator d t over denominator d x end fraction equals 1 minus t squared space divided by times d x space divided by colon open parentheses 1 minus t squared close parentheses fraction numerator begin display style d t end style over denominator begin display style 1 minus t squared end style end fraction equals d x space divided by integral integral space fraction numerator begin display style d t end style over denominator begin display style 1 minus t squared end style end fraction equals integral d x integral space fraction numerator begin display style d t end style over denominator begin display style negative open parentheses negative 1 plus t squared close parentheses end style end fraction equals x plus C subscript 1 minus integral space fraction numerator begin display style d t end style over denominator begin display style t squared minus 1 end style end fraction equals x plus C subscript 1 minus 1 half ln open vertical bar fraction numerator t minus 1 over denominator t plus 1 end fraction close vertical bar equals x plus C subscript 1 space divided by times open parentheses negative 2 close parentheses ln open vertical bar fraction numerator t minus 1 over denominator t plus 1 end fraction close vertical bar equals negative 2 x plus C subscript 1 open vertical bar fraction numerator t minus 1 over denominator t plus 1 end fraction close vertical bar equals e to the power of negative 2 x plus C subscript 1 end exponent fraction numerator t minus 1 over denominator t plus 1 end fraction equals C subscript 1 e to the power of negative 2 x end exponent space divided by times open parentheses t plus 1 close parentheses t minus 1 equals C subscript 1 t e to the power of negative 2 x end exponent plus C subscript 1 e to the power of negative 2 x end exponent t minus C subscript 1 t e to the power of negative 2 x end exponent equals 1 plus C subscript 1 e to the power of negative 2 x end exponent t open parentheses 1 minus C subscript 1 e to the power of negative 2 x end exponent close parentheses equals 1 plus C subscript 1 e to the power of negative 2 x end exponent t equals fraction numerator 1 plus C subscript 1 e to the power of negative 2 x end exponent over denominator 1 minus C subscript 1 e to the power of negative 2 x end exponent end fraction t equals fraction numerator 1 plus C subscript 1 over e to the power of 2 x end exponent over denominator 1 minus C subscript 1 over e to the power of 2 x end exponent end fraction t equals fraction numerator e to the power of 2 x end exponent over e to the power of 2 x end exponent plus C subscript 1 over e to the power of 2 x end exponent over denominator e to the power of 2 x end exponent over e to the power of 2 x end exponent minus C subscript 1 over e to the power of 2 x end exponent end fraction t equals fraction numerator fraction numerator e to the power of 2 x end exponent plus C subscript 1 over denominator e to the power of 2 x end exponent end fraction over denominator fraction numerator e to the power of 2 x end exponent minus C subscript 1 over denominator e to the power of 2 x end exponent end fraction end fraction t equals fraction numerator e to the power of 2 x end exponent plus C subscript 1 over denominator e to the power of 2 x end exponent end fraction times fraction numerator e to the power of 2 x end exponent over denominator e to the power of 2 x end exponent minus C subscript 1 end fraction t equals fraction numerator e to the power of 2 x end exponent plus C subscript 1 over denominator e to the power of 2 x end exponent minus C subscript 1 end fraction$

Wracamy teraz do podstawienia:

$y apostrophe equals fraction numerator e to the power of 2 x end exponent plus C subscript 1 over denominator e to the power of 2 x end exponent minus C subscript 1 end fraction y apostrophe equals fraction numerator e to the power of 2 x end exponent plus C subscript 1 over denominator e to the power of 2 x end exponent minus C subscript 1 end fraction space divided by integral y equals integral fraction numerator e to the power of 2 x end exponent plus C subscript 1 over denominator e to the power of 2 x end exponent minus C subscript 1 end fraction d x$

No i rozwiązujemy tą całeczkę:

$integral fraction numerator e to the power of 2 x end exponent plus C subscript 1 over denominator e to the power of 2 x end exponent minus C subscript 1 end fraction d x equals integral fraction numerator open parentheses e to the power of x close parentheses squared plus C subscript 1 over denominator open parentheses e to the power of x close parentheses squared minus C subscript 1 end fraction d x equals open vertical bar table row cell t equals e to the power of x end cell row cell d t equals e to the power of x d x end cell row cell fraction numerator d t over denominator t end fraction equals d x end cell end table close vertical bar equals integral fraction numerator t squared plus C subscript 1 over denominator t squared minus C subscript 1 end fraction fraction numerator d t over denominator t end fraction equals equals integral fraction numerator t squared plus C subscript 1 over denominator t open parentheses t squared minus C subscript 1 close parentheses end fraction d t$

Jest to całka wymierna:

$integral fraction numerator t squared plus C subscript 1 over denominator t open parentheses t squared minus C subscript 1 close parentheses end fraction d t equals integral fraction numerator t squared plus C subscript 1 over denominator t open parentheses t minus square root of C subscript 1 end root close parentheses open parentheses t plus square root of C subscript 1 end root close parentheses end fraction d t equals... fraction numerator t squared plus C subscript 1 over denominator t open parentheses t minus square root of C subscript 1 end root close parentheses open parentheses t plus square root of C subscript 1 end root close parentheses end fraction equals A over t plus fraction numerator B over denominator t minus square root of C subscript 1 end root end fraction plus fraction numerator C over denominator t plus square root of C subscript 1 end root end fraction space divided by times t open parentheses t minus square root of C subscript 1 end root close parentheses open parentheses t plus square root of C subscript 1 end root close parentheses t squared plus C subscript 1 equals A open parentheses t minus square root of C subscript 1 end root close parentheses open parentheses t plus square root of C subscript 1 end root close parentheses plus B t open parentheses t plus square root of C subscript 1 end root close parentheses plus C t open parentheses t minus square root of C subscript 1 end root close parentheses t squared plus C subscript 1 equals A t squared minus A C subscript 1 plus B t squared plus B t square root of C subscript 1 end root plus C t squared minus C t square root of C subscript 1 end root open curly brackets table row cell 1 equals A plus B plus C end cell row cell 0 equals B square root of C subscript 1 end root minus C square root of C subscript 1 end root end cell row cell C subscript 1 equals negative A C subscript 1 space divided by colon C subscript 1 comma space z a ł. space C subscript 1 not equal to 0 end cell end table close 1 equals negative A A equals negative 1 open curly brackets table attributes columnalign left end attributes row cell 1 equals negative 1 plus B plus C end cell row cell 0 equals B square root of C subscript 1 end root minus C square root of C subscript 1 end root end cell end table close open curly brackets table attributes columnalign left end attributes row cell 2 equals B plus C end cell row cell C square root of C subscript 1 end root equals B square root of C subscript 1 end root space divided by colon square root of C subscript 1 end root end cell end table close B equals C 2 equals C plus C 2 C equals 2 C equals 1 B equals 1 ... equals integral open parentheses fraction numerator negative 1 over denominator t end fraction plus fraction numerator 1 over denominator t minus square root of C subscript 1 end root end fraction plus fraction numerator 1 over denominator t plus square root of C subscript 1 end root end fraction close parentheses d t equals negative integral 1 over t d t plus integral fraction numerator 1 over denominator t minus square root of C subscript 1 end root end fraction d t plus plus integral fraction numerator 1 over denominator t plus square root of C subscript 1 end root end fraction d t equals negative ln open vertical bar t close vertical bar plus ln open vertical bar t minus square root of C subscript 1 end root close vertical bar plus ln open vertical bar t plus square root of C subscript 1 end root close vertical bar plus C subscript 2 equals equals negative ln open vertical bar e to the power of x close vertical bar plus ln open vertical bar e to the power of x minus square root of C subscript 1 end root close vertical bar plus ln open vertical bar e to the power of x plus square root of C subscript 1 end root close vertical bar plus C subscript 2 equals equals negative ln e to the power of x plus ln open vertical bar e to the power of x minus square root of C subscript 1 end root close vertical bar open vertical bar e to the power of x plus square root of C subscript 1 end root close vertical bar plus C subscript 2 equals negative x plus ln open vertical bar e to the power of 2 x end exponent minus C subscript 1 close vertical bar plus C subscript 2$

Krystian

w poście:

"Kalkulator do równań różniczkowych"

Sposób rozwiązywania tego typu całek przedstawiłem w moim Kursie Całek Nieoznaczonych.

Pójdzie tak:

$integral fraction numerator 2 x plus 7 over denominator x cubed minus 16 x end fraction d x equals integral fraction numerator begin display style 2 x plus 7 end style over denominator begin display style x open parentheses x squared minus 16 close parentheses end style end fraction d x equals integral fraction numerator begin display style 2 x plus 7 end style over denominator begin display style x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses end style end fraction d x equals...$

$fraction numerator 2 x plus 7 over denominator x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses end fraction equals A over x plus fraction numerator B over denominator x minus 4 end fraction plus fraction numerator C over denominator x plus 4 end fraction space space space divided by times x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses 2 x plus 7 equals A open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses plus B x open parentheses x plus 4 close parentheses plus C x open parentheses x minus 4 close parentheses 2 x plus 7 equals A x squared minus 16 A plus B x squared plus 4 B x plus C x squared minus 4 C x open curly brackets table row cell 0 equals A plus B plus C end cell row cell 2 equals 4 B minus 4 C end cell row cell 7 equals negative 16 A end cell end table close$

$open curly brackets table row cell 0 equals A plus B plus C end cell row cell 2 equals 4 B minus 4 C space space divided by colon 4 end cell row cell A equals negative 7 over 16 end cell end table close open curly brackets table attributes columnalign left end attributes row cell 0 equals negative 7 over 16 plus B plus C end cell row cell 1 half equals B minus C space space rightwards double arrow space space B equals 1 half plus C end cell end table close 0 equals negative 7 over 16 plus 1 half plus C plus C space space divided by times 16 0 equals negative 7 plus 8 plus 16 C plus 16 C 32 C equals negative 1 C equals negative 1 over 32$

$B equals 1 half minus 1 over 32 equals 15 over 32$

$... equals integral fraction numerator begin display style 2 x plus 7 end style over denominator begin display style x open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses end style end fraction d x equals integral fraction numerator negative 7 over 16 over denominator x end fraction d x plus integral fraction numerator 15 over 32 over denominator x minus 4 end fraction d x plus integral fraction numerator negative 1 over 32 over denominator x plus 4 end fraction d x equals equals negative 7 over 16 integral 1 over x d x plus 15 over 32 integral fraction numerator 1 over denominator x minus 4 end fraction d x minus 1 over 32 integral fraction numerator 1 over denominator x plus 4 end fraction d x equals equals negative 7 over 16 ln open vertical bar x close vertical bar plus 15 over 32 ln open vertical bar x minus 4 close vertical bar minus 1 over 32 ln open vertical bar x plus 4 close vertical bar plus C$