Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`("d"y)/("d"x) - xsqrt(25 - x^2)` = 0
Advertisements
उत्तर
The equation can be written as
`("d"y)/("d"x) - xsqrt(25 - x^2)` .........(1)
Take 25 – x2 = t
– 2x dx = dt
x dx = `- "dt"/2`
Substituting these values in equation (1), we get
dy = `xsqrt(25 - x^2) "d"x`
dy = `- sqrt("t") "dt"/2`
Taking integration on both sides, we get
`int "d"y = - "dt"/2 int "t"^(1/2) "dt"`
y = `- 1/2 ("t"^(1/2 + 1))/(1/2 + 1) + "C"`
= `- 1/2 "t"^(3/2)/(3/2) + "C"`
= `- 1/2 xx 2/3 "t"^(3/2) + "C"`
= `- 1/3 "t"^(3/2) + "C"`
y = `(-"t"^(3/2) + 3"C")/3`
3y = `-"t"^(3/2) + 3"C"`
3y = `-(25 - x^2)^(3/2) + 3"C"` .......(∵ t = 25 – x2)
APPEARS IN
संबंधित प्रश्न
The velocity v, of a parachute falling vertically satisfies the equation `"v" (dv)/(dx) = "g"(1 - v^2/k^2)` where g and k are constants. If v and are both initially zero, find v in terms of x
Find the equation of the curve whose slope is `(y - 1)/(x^2 + x)` and which passes through the point (1, 0)
Solve the following differential equation:
`("d"y)/("d"x) = sqrt((1 - y^2)/(1 - x^2)`
Solve the following differential equation:
`y"d"x + (1 + x^2)tan^-1x "d"y`= 0
Solve the following differential equation:
`("d"y)/("d"x) = "e"^(x + y) - x^3"e"^y`
Solve the following differential equation:
`2xy"d"x + (x^2 + 2y^2)"d"y` = 0
Solve the following differential equation:
`(y^2 - 2xy) "d"x = (x^2 - 2xy) "d"y`
Solve the following differential equation:
`x ("d"y)/("d"x) = y - xcos^2(y/x)`
Choose the correct alternative:
The solution of `("d"y)/("d"x) = 2^(y - x)` is
Solve : cos x(1 + cosy) dx – sin y(1 + sinx) dy = 0
Solve: `log(("d"y)/("d"x))` = ax + by
Solve the following homogeneous differential equation:
`(x - y) ("d"y)/("d"x) = x + 3y`
Solve the following:
`("d"y)/(""dx) + y cos x = sin x cos x`
Solve the following:
`("d"y)/("d"x) + (3x^2)/(1 + x^3) y = (1 + x^2)/(1 + x^3)`
Choose the correct alternative:
The differential equation of y = mx + c is (m and c are arbitrary constants)
Choose the correct alternative:
Solution of `("d"x)/("d"y) + "P"x = 0`
Choose the correct alternative:
If sec2 x is an integrating factor of the differential equation `("d"y)/("d"x) + "P"y` = Q then P =
Choose the correct alternative:
The solution of the differential equation `("d"y)/("d"x) + "P"y` = Q where P and Q are the function of x is
Choose the correct alternative:
The solution of the differential equation `("d"y)/("d"x) = y/x + (f(y/x))/(f"'"(y/x))` is
