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Answered on 06 Apr Learn Integrals
Sadika
To find the antiderivative of the function 3x2+4x33x2+4x3, we'll integrate each term separately using the power rule of integration.
For 3x23x2: ∫3x2 dx=3⋅x33+C=x3+C∫3x2dx=3⋅3x3+C=x3+C
For 4x34x3: ∫4x3 dx=4⋅x44+C=x4+C∫4x3dx=4⋅4x4+C=x4+C
Putting them together, the antiderivative of 3x2+4x33x2+4x3 is: ∫(3x2+4x3) dx=x3+x4+C∫(3x2+4x3)dx=x3+x4+C
where CC is the constant of integration.
Answered on 06 Apr Learn Integrals
Sadika
To find the antiderivative F(x)F(x) of the function f(x)=4x3−6f(x)=4x3−6, we integrate each term separately using the power rule of integration.
For 4x34x3, we integrate term-by-term:
∫4x3 dx=4⋅x44+C1=x4+C1∫4x3dx=4⋅4x4+C1=x4+C1
For −6−6, we integrate it as a constant:
∫(−6) dx=−6x+C2∫(−6)dx=−6x+C2
Now, combining the results, we get:
F(x)=x4−6x+CF(x)=x4−6x+C
where C=C1+C2C=C1+C2 is the constant of integration.
Given that F(0)=3F(0)=3, we can find the value of the constant CC:
F(0)=04−6(0)+C=3F(0)=04−6(0)+C=3 C=3C=3
Therefore, the antiderivative F(x)F(x) is:
F(x)=x4−6x+3F(x)=x4−6x+3
Answered on 06 Apr Learn Integrals
Sadika
To integrate the function 2xsin(x2+1)2xsin(x2+1) with respect to xx using integration by substitution, let's make the substitution:
u=x2+1u=x2+1
Then, we find dudu:
dudx=2xdxdu=2x
Now, we can express dxdx in terms of dudu:
dx=du2xdx=2xdu
Substituting uu and dxdx into the integral, we have:
∫2xsin(x2+1) dx=∫sin(u)⋅du∫2xsin(x2+1)dx=∫sin(u)⋅du
Now, we integrate sin(u)sin(u) with respect to uu:
∫sin(u) du=−cos(u)+C∫sin(u)du=−cos(u)+C
Now, we need to substitute back u=x2+1u=x2+1:
−cos(x2+1)+C−cos(x2+1)+C
So, the antiderivative of 2xsin(x2+1)2xsin(x2+1) with respect to xx is:
−cos(x2+1)+C−cos(x2+1)+C
where CC is the constant of integration.
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Answered on 06 Apr Learn Differential Equations
Sadika
To form the differential equation representing the family of curves y=asin(x+b)y=asin(x+b), where aa and bb are arbitrary constants, we need to find the first and second derivatives of yy with respect to xx.
Given function: y=asin(x+b)y=asin(x+b)
First, let's find the first derivative of yy with respect to xx:
dydx=acos(x+b)dxdy=acos(x+b)
Now, let's find the second derivative of yy with respect to xx:
d2ydx2=−asin(x+b)dx2d2y=−asin(x+b)
Now, we need to express yy, dydxdxdy, and d2ydx2dx2d2y in terms of yy and xx to form the differential equation.
We have:
y=asin(x+b)y=asin(x+b)
dydx=acos(x+b)dxdy=acos(x+b)
d2ydx2=−asin(x+b)dx2d2y=−asin(x+b)
Substituting these into the differential equation d2ydx2+y=0dx2d2y+y=0, we get:
−asin(x+b)+asin(x+b)=0−asin(x+b)+asin(x+b)=0
0=00=0
This equation is trivially satisfied.
However, let's represent the differential equation without cancelling out terms:
d2ydx2+y=0dx2d2y+y=0
−asin(x+b)+asin(x+b)+asin(x+b)=0−asin(x+b)+asin(x+b)+asin(x+b)=0
asin(x+b)−asin(x+b)=0asin(x+b)−asin(x+b)=0
0=00=0
This differential equation represents the family of curves y=asin(x+b)y=asin(x+b), where aa and bb are arbitrary constants.
Answered on 06 Apr Learn Differential Equations
Sadika
The family of lines passing through the origin has the general form y=mxy=mx, where mm is the slope of the line. To find the differential equation representing this family of lines, we need to express the derivative of yy with respect to xx in terms of yy and xx, which will give us the differential equation.
Given the family of lines y=mxy=mx, where mm is the slope, we differentiate yy with respect to xx:
dydx=mdxdy=m
This derivative represents the slope of the line at any point (x,y)(x,y). Now, since y=mxy=mx, we have m=yxm=xy.
Substituting this into dydx=mdxdy=m, we get:
dydx=yxdxdy=xy
This is the differential equation representing the family of lines passing through the origin.
Answered on 08 Apr Learn Differential Equations
Sadika
To form the differential equation of the family of circles having a center on the y-axis and a radius of 3 units, we first need to express the equation of a circle with its center on the y-axis and radius 3 units.
The general equation of a circle with center (0, c) and radius r is given by:
x2+(y−c)2=r2x2+(y−c)2=r2
In this case, since the center lies on the y-axis, the x-coordinate of the center is 0, and the y-coordinate can be any value (let's denote it as cc).
Substituting cc with yy and rr with 33, we get:
x2+(y−y)2=32x2+(y−y)2=32 x2+y2=9x2+y2=9
Now, to form the differential equation, we'll differentiate both sides with respect to xx:
ddx(x2)+ddx(y2)=ddx(9)dxd(x2)+dxd(y2)=dxd(9) 2x+2ydydx=02x+2ydxdy=0
This can be simplified to:
x+ydydx=0x+ydxdy=0
And finally, solving for dydxdxdy, we get the differential equation:
dydx=−xydxdy=−yx
This is the differential equation of the family of circles with centers on the y-axis and radius 3 units.
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Answered on 08 Apr Learn Differential Equations
Sadika
To find the general solution of the given differential equation:
dydx=1+y21+x2dxdy=1+1+x2y2
Let's rewrite the equation in a more suitable form:
dydx=11+y21+x2dxdy=11+1+x2y2
Now, this looks like a separable differential equation. We can rewrite it as:
dydx=11+y21+x2dxdy=11+1+x2y2 dydx=1+x2+y21+x2dxdy=1+x21+x2+y2
Now, we can separate variables:
1+x21+x2+y2dy=dx1+x2+y21+x2dy=dx
Integrating both sides:
∫1+x21+x2+y2 dy=∫dx∫1+x2+y21+x2dy=∫dx
Let's denote u=1+x2+y2u=1+x2+y2, then du=2y dydu=2ydy:
12∫1u du=x+C21∫u1du=x+C
12ln∣u∣=x+C21ln∣u∣=x+C
Substitute back u=1+x2+y2u=1+x2+y2:
12ln∣1+x2+y2∣=x+C21ln∣1+x2+y2∣=x+C
ln∣1+x2+y2∣=2x+2Cln∣1+x2+y2∣=2x+2C
∣1+x2+y2∣=e2x+2C∣1+x2+y2∣=e2x+2C
1+x2+y2=Ae2x1+x2+y2=Ae2x
Where A=±e2CA=±e2C.
Finally, if we let A=e2CA=e2C, we can express the solution explicitly:
1+x2+y2=e2x1+x2+y2=e2x
y2=e2x−1−x2y2=e2x−1−x2
y=±e2x−1−x2y=±e2x−1−x2
So, the general solution of the given differential equation is:
y=±e2x−1−x2y=±e2x−1−x2
Answered on 08 Apr Learn Differential Equations
Sadika
Answered on 08 Apr Learn Differential Equations
Sadika
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Answered on 08 Apr Learn Differential Equations
Sadika
To find the integrating factor of the given differential equation (1−x2)dydx−xy=1(1−x2)dxdy−xy=1, we can use the formula for the integrating factor μ(x)μ(x), which is given by:
where P(x)P(x) is the coefficient of yy in the given differential equation.
In this case, P(x)=−xP(x)=−x.
So,
Therefore, the integrating factor μ(x)μ(x) is:
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