Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation x²-64=0
-
Equation log2(x^2+3x)=2
- Equation log7(3-x)=2log7(4)
- Equation (D²-2D+1)y=cos3x
- Express {x} in terms of y where:
- (x^2+y^2-1)^3+x^2*y^3=0
- -1*x-19*y=-10
- 5*x+5*y=20
- 4*x-16*y=6
- Derivative of:
-
cos3x
- Integral of d{x}:
-
cos3x
- Graphing y =:
- cos3x
-
Identical expressions
- (D²-2D+ one)y=cos3x
- (D² minus 2D plus 1)y equally co sinus of e of 3x
- (D² minus 2D plus one)y equally co sinus of e of 3x
- D²-2D+1y=cos3x
-
Similar expressions
- (D²+2D+1)y=cos3x
- sin(6x)-cos(3x)=0
- 2cos^3x-2cosx-sin^2x=0
- (D²-2D-1)y=cos3x
- sin^3x+cos^3x=sin^2x+cos^2x
(D²-2D+1)y=cos3x equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/ 2 \ \d - 2*d + 1/*y = cos(3*x)
$$y \left(\left(d^{2} - 2 d\right) + 1\right) = \cos{\left(3 x \right)}$$
Detail solution
Given the equation
$$y \left(\left(d^{2} - 2 d\right) + 1\right) = \cos{\left(3 x \right)}$$
transform
$$y \left(d^{2} - 2 d + 1\right) - \cos{\left(3 x \right)} - 1 = 0$$
$$y \left(\left(d^{2} - 2 d\right) + 1\right) - \cos{\left(3 x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(3 x \right)}$$
Expand brackets in the left part
Looking for similar summands in the left part:
Move free summands (without w)
from left part to right part, we given:
$$- w + y \left(d^{2} - 2 d + 1\right) = 1$$
Divide both parts of the equation by (-w + y*(1 + d^2 - 2*d))/w
We get the answer: w = -1 + y + y*d^2 - 2*d*y
do backward replacement
$$\cos{\left(3 x \right)} = w$$
substitute w:
$$y \left(\left(d^{2} - 2 d\right) + 1\right) = \cos{\left(3 x \right)}$$
transform
$$y \left(d^{2} - 2 d + 1\right) - \cos{\left(3 x \right)} - 1 = 0$$
$$y \left(\left(d^{2} - 2 d\right) + 1\right) - \cos{\left(3 x \right)} - 1 = 0$$
Do replacement
$$w = \cos{\left(3 x \right)}$$
Expand brackets in the left part
-1 - w + d+2+2*d+1y = 0
Looking for similar summands in the left part:
-1 - w + y*(1 + d^2 - 2*d) = 0
Move free summands (without w)
from left part to right part, we given:
$$- w + y \left(d^{2} - 2 d + 1\right) = 1$$
Divide both parts of the equation by (-w + y*(1 + d^2 - 2*d))/w
w = 1 / ((-w + y*(1 + d^2 - 2*d))/w)
We get the answer: w = -1 + y + y*d^2 - 2*d*y
do backward replacement
$$\cos{\left(3 x \right)} = w$$
substitute w:
The solution of the parametric equation
Given the equation with a parameter:
$$y \left(d^{2} - 2 d + 1\right) = \cos{\left(3 x \right)}$$
Коэффициент при y равен
$$d^{2} - 2 d + 1$$
then possible cases for d :
$$d < 1$$
$$d = 1$$
Consider all cases in more detail:
With
$$d < 1$$
the equation
$$y - \cos{\left(3 x \right)} = 0$$
its solution
$$y = \cos{\left(3 x \right)}$$
With
$$d = 1$$
the equation
$$- \cos{\left(3 x \right)} = 0$$
its solution
$$y \left(d^{2} - 2 d + 1\right) = \cos{\left(3 x \right)}$$
Коэффициент при y равен
$$d^{2} - 2 d + 1$$
then possible cases for d :
$$d < 1$$
$$d = 1$$
Consider all cases in more detail:
With
$$d < 1$$
the equation
$$y - \cos{\left(3 x \right)} = 0$$
its solution
$$y = \cos{\left(3 x \right)}$$
With
$$d = 1$$
the equation
$$- \cos{\left(3 x \right)} = 0$$
its solution
The graph
Rapid solution
[src]
/ cos(3*x) \ / cos(3*x) \
y1 = I*im|------------| + re|------------|
| 2 | | 2 |
\1 + d - 2*d/ \1 + d - 2*d/
$$y_{1} = \operatorname{re}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)} + i \operatorname{im}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)}$$
y1 = re(cos(3*x)/(d^2 - 2*d + 1)) + i*im(cos(3*x)/(d^2 - 2*d + 1))
Sum and product of roots
[src]
sum
/ cos(3*x) \ / cos(3*x) \
I*im|------------| + re|------------|
| 2 | | 2 |
\1 + d - 2*d/ \1 + d - 2*d/
$$\operatorname{re}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)} + i \operatorname{im}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)}$$
=
/ cos(3*x) \ / cos(3*x) \
I*im|------------| + re|------------|
| 2 | | 2 |
\1 + d - 2*d/ \1 + d - 2*d/
$$\operatorname{re}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)} + i \operatorname{im}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)}$$
product
/ cos(3*x) \ / cos(3*x) \
I*im|------------| + re|------------|
| 2 | | 2 |
\1 + d - 2*d/ \1 + d - 2*d/
$$\operatorname{re}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)} + i \operatorname{im}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)}$$
=
/ cos(3*x) \ / cos(3*x) \
I*im|------------| + re|------------|
| 2 | | 2 |
\1 + d - 2*d/ \1 + d - 2*d/
$$\operatorname{re}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)} + i \operatorname{im}{\left(\frac{\cos{\left(3 x \right)}}{d^{2} - 2 d + 1}\right)}$$
i*im(cos(3*x)/(1 + d^2 - 2*d)) + re(cos(3*x)/(1 + d^2 - 2*d))