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Question
Solve the following differential equation:
`("d"y)/("d"x) - xsqrt(25 - x^2)` = 0
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Solution
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)
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