Question

Solve the following differential equation:

`("d"y)/("d"x) - xsqrt(25 - x^2)` = 0

Answer 1

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)