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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

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