Question

Let a curve y = y (x) be given by the solution of differential equation cos(1/2cos⁻¹(e⁻ˣ))dx=√e²ˣ−1dy If it intersects y – axis at y = –1, and the intersection point point of the curve with x – axis is (α,θ), then eα is equal to __________.

Answer 1

Let a curve y = y (x) be given by the solution of differential equation cos(1/2cos⁻¹(e⁻ˣ))dx=√e²ˣ−1dy If it intersects y – axis at y = –1, and the intersection point point of the curve with x – axis is (α,0), then eα is equal to
`e^\alpha=1`

Let's solve the given differential equation and find the curve

y=y(x).

The differential equation is given by:

`cos((1)/(2)cos^(-1)(e^(-x)) )dx = \sqrt{e^(2x)-1} \:dy`

To solve this, we'll separate variables and integrate:

`\int cos((1)/(2)\: cos^(-1)(e^(-x)) )dx = \int \sqrt{e^(2x)-1} \:dy`

Let's denote
`u = (1)/(2) \: cos^(-1) (e^(-x))`

cos(2u)=2
`cos^2` (u)−1 and sin(2u)=2sin(u)cos(u). The equation becomes:

`\int cos(2u)\:du=\int \sqrt{e^(2x)-1} \: dy`

`(1)/(2) \int (2cos^2(u)-1)\:du=\int \sqrt{e^(2x)-1} \: dy`

`\int (cos^2(u)-sin^2(u))\:du=\int \sqrt{e^(2x)-1} \: dy`

`\int (cos(2u))\:du=\int \sqrt{e^(2x)-1} \: dy`

`(1)/(2)sin(2u)= \int \sqrt{e^(2x)-1} \: dy`

`sin(2u)= 2\int \sqrt{e^(2x)-1} \: dy`

Now, we can integrate both sides:

`-cos(2u) = 2\int\sqrt{e^(2x)-1} \:dy`

`cos(2u) = -2\int\sqrt{e^(2x)-1} \:dy`

Now, substitute back
`u = (1)/(2) \:(cos^(-1)(e^(-x))`

`cos(cos^(-1)(e^(-x)) = -2\int\sqrt{e^(2x)-1} \:dy`

`e^(-x)= -2\int\sqrt{e^(2x)-1} \:dy`

Now, let's solve for y(x). The curve intersects the y-axis at y = −1, which means when x=0:

`e^0=-2\int√(e^0-1)\:dy`

`1=-2\int 0 \:dy`

This implies that the constant of integration is 1. So, the curve is given by:

`e^(-x)= -2\int\sqrt{e^(2x)-1} \:dy+1`

Now, to find
`e^\alpha`, where (α,0) is the intersection point with the x-axis, substitute y=0

`e^(-\alpha) = -2 \int \sqrt{e^(2\alpha)-1} \:(0) +1 dy`

`e^(-\alpha) =1`

So,
`e^(\alpha) =1`

Question:-

Let a curve y = y (x) be given by the solution of differential equation cos(1/2cos⁻¹(e⁻ˣ))dx=√e²ˣ−1dy If it intersects y – axis at y = –1, and the intersection point point of the curve with x – axis is (α,o), then eα is equal to __________.