Sr Examen
Ecuación diferencial sqrt(3+y^2)*dx=sqrt((x^2)*y-y)*dy
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Solución
Ha introducido
[src]
___________ _________________
/ 2 / 2 d
\/ 3 + y (x) = \/ -y(x) + x *y(x) *--(y(x))
dx
$$\sqrt{y^{2}{\left(x \right)} + 3} = \sqrt{x^{2} y{\left(x \right)} - y{\left(x \right)}} \frac{d}{d x} y{\left(x \right)}$$
sqrt(y^2 + 3) = sqrt(x^2*y - y)*y'
Respuesta
[src]
// / // -I*log(y(x)) for |y(x)| < 1\ \
|| | || | |
|| | || / 1 \ 1 | |
|| | || I*log|----| for ------ < 1| |
|| | |< \y(x)/ |y(x)| |*asin(x) |
|| | || | _ / | 2 pi*I\ |
|| | || __2, 0 / 1, 1 | \ __0, 2 /1, 1 | \ | ___ 3/2 |_ |1/2, 3/4 | y (x)*e | |
|| | ||I*/__ | | y(x)| - I*/__ | | y(x)| otherwise | \/ 3 *y (x)*Gamma(3/4)* | | | -----------| |
|| |acosh(x)*log(y(x)) \\ \_|2, 2 \0, 0 | / \_|2, 2 \ 0, 0 | / / 2 1 \ 7/4 | 3 / |
|| <------------------ - ---------------------------------------------------------------------------------------- + ------------------------------------------------------ for y(x) < 0 for Or(And(x >= -oo, x < -1), And(x > 1, x < oo))|
|| | 2 2 6*Gamma(7/4) |
|| | |
|| | |
|| | _ / | 2 pi*I\ |
|| | ___ 3/2 |_ |1/2, 3/4 | y (x)*e | |
// acosh(x) | 2| \ || | \/ 3 *y (x)*Gamma(3/4)* | | | -----------| |
|| -------- for |x | > 1| || | 2 1 \ 7/4 | 3 / |
|| ______ | || | zoo*acosh(x) + zoo*asin(x) + ------------------------------------------------------ otherwise |
______ || \/ y(x) | || \ 6*Gamma(7/4) |
- \/ y(x) *|< | + |< | = C1
||-I*asin(x) | ||/ // -I*log(y(x)) for |y(x)| < 1\ |
||----------- otherwise | ||| || | |
|| ______ | ||| || / 1 \ 1 | |
\\ \/ y(x) / ||| || I*log|----| for ------ < 1| |
||| |< \y(x)/ |y(x)| |*asin(x) |
||| || | _ / | 2 pi*I\ |
||| || __2, 0 / 1, 1 | \ __0, 2 /1, 1 | \ | ___ 3/2 |_ |1/2, 3/4 | y (x)*e | |
||| ||I*/__ | | y(x)| - I*/__ | | y(x)| otherwise | \/ 3 *y (x)*Gamma(3/4)* | | | -----------| |
||| \\ \_|2, 2 \0, 0 | / \_|2, 2 \ 0, 0 | / / I*asin(x)*log(y(x)) 2 1 \ 7/4 | 3 / |
||<- ---------------------------------------------------------------------------------------- - ------------------- + ------------------------------------------------------ for y(x) < 0 otherwise |
||| 2 2 6*Gamma(7/4) |
||| |
||| |
||| _ / | 2 pi*I\ |
||| ___ 3/2 |_ |1/2, 3/4 | y (x)*e | |
||| \/ 3 *y (x)*Gamma(3/4)* | | | -----------| |
||| acosh(x)*log(y(x)) I*asin(x)*log(y(x)) 2 1 \ 7/4 | 3 / |
||| -oo*sign(I*asin(x) + acosh(x)) + zoo*asin(x) - ------------------ - ------------------- + ------------------------------------------------------ otherwise |
\\\ 2 2 6*Gamma(7/4) /
$$- \left(\begin{cases} \frac{\operatorname{acosh}{\left(x \right)}}{\sqrt{y{\left(x \right)}}} & \text{for}\: \left|{x^{2}}\right| > 1 \\- \frac{i \operatorname{asin}{\left(x \right)}}{\sqrt{y{\left(x \right)}}} & \text{otherwise} \end{cases}\right) \sqrt{y{\left(x \right)}} + \begin{cases} \begin{cases} - \frac{\left(\begin{cases} - i \log{\left(y{\left(x \right)} \right)} & \text{for}\: \left|{y{\left(x \right)}}\right| < 1 \\i \log{\left(\frac{1}{y{\left(x \right)}} \right)} & \text{for}\: \frac{1}{\left|{y{\left(x \right)}}\right|} < 1 \\i {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {y{\left(x \right)}} \right)} - i {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {y{\left(x \right)}} \right)} & \text{otherwise} \end{cases}\right) \operatorname{asin}{\left(x \right)}}{2} + \frac{\sqrt{3} y^{\frac{3}{2}}{\left(x \right)} \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {\frac{e^{i \pi} y^{2}{\left(x \right)}}{3}} \right)}}{6 \Gamma\left(\frac{7}{4}\right)} + \frac{\log{\left(y{\left(x \right)} \right)} \operatorname{acosh}{\left(x \right)}}{2} & \text{for}\: y{\left(x \right)} < 0 \\\frac{\sqrt{3} y^{\frac{3}{2}}{\left(x \right)} \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {\frac{e^{i \pi} y^{2}{\left(x \right)}}{3}} \right)}}{6 \Gamma\left(\frac{7}{4}\right)} + \tilde{\infty} \operatorname{acosh}{\left(x \right)} + \tilde{\infty} \operatorname{asin}{\left(x \right)} & \text{otherwise} \end{cases} & \text{for}\: \left(x \geq -\infty \wedge x < -1\right) \vee \left(x > 1 \wedge x < \infty\right) \\\begin{cases} - \frac{\left(\begin{cases} - i \log{\left(y{\left(x \right)} \right)} & \text{for}\: \left|{y{\left(x \right)}}\right| < 1 \\i \log{\left(\frac{1}{y{\left(x \right)}} \right)} & \text{for}\: \frac{1}{\left|{y{\left(x \right)}}\right|} < 1 \\i {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {y{\left(x \right)}} \right)} - i {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {y{\left(x \right)}} \right)} & \text{otherwise} \end{cases}\right) \operatorname{asin}{\left(x \right)}}{2} + \frac{\sqrt{3} y^{\frac{3}{2}}{\left(x \right)} \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {\frac{e^{i \pi} y^{2}{\left(x \right)}}{3}} \right)}}{6 \Gamma\left(\frac{7}{4}\right)} - \frac{i \log{\left(y{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}}{2} & \text{for}\: y{\left(x \right)} < 0 \\\frac{\sqrt{3} y^{\frac{3}{2}}{\left(x \right)} \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {\frac{e^{i \pi} y^{2}{\left(x \right)}}{3}} \right)}}{6 \Gamma\left(\frac{7}{4}\right)} - \frac{\log{\left(y{\left(x \right)} \right)} \operatorname{acosh}{\left(x \right)}}{2} - \frac{i \log{\left(y{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}}{2} + \tilde{\infty} \operatorname{asin}{\left(x \right)} - \infty \operatorname{sign}{\left(\operatorname{acosh}{\left(x \right)} + i \operatorname{asin}{\left(x \right)} \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases} = C_{1}$$
Gráfico para el problema de Cauchy
Clasificación
factorable
1st power series
lie group
Respuesta numérica
[src]
(x, y):
(0.0, 0.75)
(1.1111111111111112, nan)
(2.2222222222222223, nan)
(3.3333333333333335, nan)
(4.444444444444445, nan)
(5.555555555555555, nan)
(6.666666666666667, nan)
(7.777777777777779, nan)
(8.88888888888889, nan)
(10.0, nan)
(10.0, nan)