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Identical expressions
- y^ two *sin(x)*d+cos^ two (x)*ln(y)*d= zero
- y squared multiply by sinus of (x) multiply by d plus co sinus of e of squared (x) multiply by ln(y) multiply by d equally 0
- y to the power of two multiply by sinus of (x) multiply by d plus co sinus of e of to the power of two (x) multiply by ln(y) multiply by d equally zero
- y2*sin(x)*d+cos2(x)*ln(y)*d=0
- y2*sinx*d+cos2x*lny*d=0
- y²*sin(x)*d+cos²(x)*ln(y)*d=0
- y to the power of 2*sin(x)*d+cos to the power of 2(x)*ln(y)*d=0
- y^2sin(x)d+cos^2(x)ln(y)d=0
- y2sin(x)d+cos2(x)ln(y)d=0
- y2sinxd+cos2xlnyd=0
- y^2sinxd+cos^2xlnyd=0
- y^2*sin(x)*d+cos^2(x)*ln(y)*d=O
-
Similar expressions
- y^2*sin(x)*d-cos^2(x)*ln(y)*d=0
- y^2*sinx*d+cos^2(x)*ln(y)*d=0
y^2*sin(x)*d+cos^2(x)*ln(y)*d=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 2 y *sin(x)*d + cos (x)*log(y)*d = 0
$$d y^{2} \sin{\left(x \right)} + d \log{\left(y \right)} \cos^{2}{\left(x \right)} = 0$$
Detail solution
Given the equation
$$d y^{2} \sin{\left(x \right)} + d \log{\left(y \right)} \cos^{2}{\left(x \right)} = 0$$
transform
$$d y^{2} \sin{\left(x \right)} + d \log{\left(y \right)} \cos^{2}{\left(x \right)} - 1 = 0$$
$$d y^{2} \sin{\left(x \right)} + d \log{\left(y \right)} \cos^{2}{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = d \log{\left(y \right)}$$
$$b = 0$$
$$c = d y^{2} \sin{\left(x \right)} - 1$$
, then
The equation has two roots.
or
$$w_{1} = \frac{\sqrt{- d \left(d y^{2} \sin{\left(x \right)} - 1\right) \log{\left(y \right)}}}{d \log{\left(y \right)}}$$
$$w_{2} = - \frac{\sqrt{- d \left(d y^{2} \sin{\left(x \right)} - 1\right) \log{\left(y \right)}}}{d \log{\left(y \right)}}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
substitute w:
$$d y^{2} \sin{\left(x \right)} + d \log{\left(y \right)} \cos^{2}{\left(x \right)} = 0$$
transform
$$d y^{2} \sin{\left(x \right)} + d \log{\left(y \right)} \cos^{2}{\left(x \right)} - 1 = 0$$
$$d y^{2} \sin{\left(x \right)} + d \log{\left(y \right)} \cos^{2}{\left(x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = d \log{\left(y \right)}$$
$$b = 0$$
$$c = d y^{2} \sin{\left(x \right)} - 1$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (d*log(y)) * (-1 + d*y^2*sin(x)) = -4*d*(-1 + d*y^2*sin(x))*log(y)
The equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{\sqrt{- d \left(d y^{2} \sin{\left(x \right)} - 1\right) \log{\left(y \right)}}}{d \log{\left(y \right)}}$$
$$w_{2} = - \frac{\sqrt{- d \left(d y^{2} \sin{\left(x \right)} - 1\right) \log{\left(y \right)}}}{d \log{\left(y \right)}}$$
do backward replacement
$$\cos{\left(x \right)} = w$$
substitute w:
The graph
Rapid solution
[src]
/ / 3*I*x I*x \\ / / 3*I*x I*x \\
| | 4*I*e 4*I*e || | | 4*I*e 4*I*e ||
/ / / 3*I*x I*x \\\ -re|W|- --------------------- + ---------------------|| -re|W|- --------------------- + ---------------------|| / / / 3*I*x I*x \\\
| | | 4*I*e 4*I*e ||| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | | 4*I*e 4*I*e |||
|im|W|- --------------------- + ---------------------||| \ \ 1 + 2*e + e 1 + 2*e + e // \ \ 1 + 2*e + e 1 + 2*e + e // |im|W|- --------------------- + ---------------------|||
| | | 2*I*x 4*I*x 2*I*x 4*I*x||| -------------------------------------------------------- -------------------------------------------------------- | | | 2*I*x 4*I*x 2*I*x 4*I*x|||
| \ \ 1 + 2*e + e 1 + 2*e + e //| 2 2 | \ \ 1 + 2*e + e 1 + 2*e + e //|
y1 = cos|------------------------------------------------------|*e - I*e *sin|------------------------------------------------------|
\ 2 / \ 2 /
$$y_{1} = - i e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \sin{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)} + e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \cos{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)}$$
y1 = -i*exp(-re(LambertW(-4*i*exp(3*i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1) + 4*i*exp(i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1)))/2)*sin(im(LambertW(-4*i*exp(3*i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1) + 4*i*exp(i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1)))/2) + exp(-re(LambertW(-4*i*exp(3*i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1) + 4*i*exp(i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1)))/2)*cos(im(LambertW(-4*i*exp(3*i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1) + 4*i*exp(i*x)/(exp(4*i*x) + 2*exp(2*i*x) + 1)))/2)
Sum and product of roots
[src]
sum
/ / 3*I*x I*x \\ / / 3*I*x I*x \\
| | 4*I*e 4*I*e || | | 4*I*e 4*I*e ||
/ / / 3*I*x I*x \\\ -re|W|- --------------------- + ---------------------|| -re|W|- --------------------- + ---------------------|| / / / 3*I*x I*x \\\
| | | 4*I*e 4*I*e ||| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | | 4*I*e 4*I*e |||
|im|W|- --------------------- + ---------------------||| \ \ 1 + 2*e + e 1 + 2*e + e // \ \ 1 + 2*e + e 1 + 2*e + e // |im|W|- --------------------- + ---------------------|||
| | | 2*I*x 4*I*x 2*I*x 4*I*x||| -------------------------------------------------------- -------------------------------------------------------- | | | 2*I*x 4*I*x 2*I*x 4*I*x|||
| \ \ 1 + 2*e + e 1 + 2*e + e //| 2 2 | \ \ 1 + 2*e + e 1 + 2*e + e //|
cos|------------------------------------------------------|*e - I*e *sin|------------------------------------------------------|
\ 2 / \ 2 /
$$- i e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \sin{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)} + e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \cos{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)}$$
=
/ / 3*I*x I*x \\ / / 3*I*x I*x \\
| | 4*I*e 4*I*e || | | 4*I*e 4*I*e ||
/ / / 3*I*x I*x \\\ -re|W|- --------------------- + ---------------------|| -re|W|- --------------------- + ---------------------|| / / / 3*I*x I*x \\\
| | | 4*I*e 4*I*e ||| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | | 4*I*e 4*I*e |||
|im|W|- --------------------- + ---------------------||| \ \ 1 + 2*e + e 1 + 2*e + e // \ \ 1 + 2*e + e 1 + 2*e + e // |im|W|- --------------------- + ---------------------|||
| | | 2*I*x 4*I*x 2*I*x 4*I*x||| -------------------------------------------------------- -------------------------------------------------------- | | | 2*I*x 4*I*x 2*I*x 4*I*x|||
| \ \ 1 + 2*e + e 1 + 2*e + e //| 2 2 | \ \ 1 + 2*e + e 1 + 2*e + e //|
cos|------------------------------------------------------|*e - I*e *sin|------------------------------------------------------|
\ 2 / \ 2 /
$$- i e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \sin{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)} + e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \cos{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)}$$
product
/ / 3*I*x I*x \\ / / 3*I*x I*x \\
| | 4*I*e 4*I*e || | | 4*I*e 4*I*e ||
/ / / 3*I*x I*x \\\ -re|W|- --------------------- + ---------------------|| -re|W|- --------------------- + ---------------------|| / / / 3*I*x I*x \\\
| | | 4*I*e 4*I*e ||| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | 2*I*x 4*I*x 2*I*x 4*I*x|| | | | 4*I*e 4*I*e |||
|im|W|- --------------------- + ---------------------||| \ \ 1 + 2*e + e 1 + 2*e + e // \ \ 1 + 2*e + e 1 + 2*e + e // |im|W|- --------------------- + ---------------------|||
| | | 2*I*x 4*I*x 2*I*x 4*I*x||| -------------------------------------------------------- -------------------------------------------------------- | | | 2*I*x 4*I*x 2*I*x 4*I*x|||
| \ \ 1 + 2*e + e 1 + 2*e + e //| 2 2 | \ \ 1 + 2*e + e 1 + 2*e + e //|
cos|------------------------------------------------------|*e - I*e *sin|------------------------------------------------------|
\ 2 / \ 2 /
$$- i e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \sin{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)} + e^{- \frac{\operatorname{re}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}} \cos{\left(\frac{\operatorname{im}{\left(W\left(- \frac{4 i e^{3 i x}}{e^{4 i x} + 2 e^{2 i x} + 1} + \frac{4 i e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} \right)}$$
=
/ / / 2*I*x\ I*x\\ / / / 2*I*x\ I*x\\
| |4*I*\1 - e /*e || | |4*I*\1 - e /*e ||
re|W|---------------------|| I*im|W|---------------------||
| | 2*I*x 4*I*x|| | | 2*I*x 4*I*x||
\ \1 + 2*e + e // \ \1 + 2*e + e //
- ---------------------------- - ------------------------------
2 2
e
$$e^{- \frac{\operatorname{re}{\left(W\left(\frac{4 i \left(1 - e^{2 i x}\right) e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2} - \frac{i \operatorname{im}{\left(W\left(\frac{4 i \left(1 - e^{2 i x}\right) e^{i x}}{e^{4 i x} + 2 e^{2 i x} + 1}\right)\right)}}{2}}$$
exp(-re(LambertW(4*i*(1 - exp(2*i*x))*exp(i*x)/(1 + 2*exp(2*i*x) + exp(4*i*x))))/2 - i*im(LambertW(4*i*(1 - exp(2*i*x))*exp(i*x)/(1 + 2*exp(2*i*x) + exp(4*i*x))))/2)