Solve the following differential equation y2dx + (xy + x2) dy = 0 - Mathematics and Statistics

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Sum

Solve the following differential equation y2dx + (xy + x2) dy = 0

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Solution

y2dx + (xy + x2) dy = 0

∴ `y^2 + (xy + x^2)  ("d"y)/("d"x)` = 0

∴ `(xy + x^2)  ("d"y)/("d"x)` = − y2

∴ `("d"y)/("d"x) = (-y^2)/(xy + x^2)`   .....(i)

Put y = tx      ......(ii)

Differentiating w.r.t. x, we get

`("d"y)/("d"x) = "t" + x  "dt"/("d"x)`  ......(iii)

Substituting (ii) and (iii) in (i), we get

`"t" + x  "dt"/("d"x) = (-"t"^2x^2)/(x("t"x) + x^2)`

∴ `"t" + x  "dt"/("d"x) = (-"t"^2x^2)/(x^2("t" + 1))`

∴ `"t" + x  "dt"/("d"x) = (-"t"^2)/(1 + "t")`

∴ `x  "dt"/("d"x) = (-"t"^2)/(1 + "t") - "t"`

= `(-"t"^2 - "t" - "t"^2)/(1 + "t")`

= `(-2"t"^2 - "t")/(1 + "t")`

∴ `(1 + "t")/("t"^2 + "t")  "dt" = -  ("d"x)/x`

Integrating on both sides, we get

`int  (1 + "t")/(2"t"^2 + "t")  "dt" = - int ("d"x)/x`

∴ `int ((2"t" + 1) - "t")/("t"(21"t" + 1))  "dt" = -int ("d"x)/x`

∴ `int (1/"t" - 1/(2"t" + 1))  "dt" = -int ("d"x)/x`

∴ `int 1/"t" "dt" - 1/2 int 2/(2"t" + 1)  "dt" = -int ("d"x)/x`

∴ `log |"t"| - 1/2 log|2"t" + 1|` = − log|x| + log |c|

∴ `log |y/x| - 1/2 log|2(y/x) + 1|` = − log|x| + log |c|

∴ `log |y| - log |x| - 1/2 log|(2y + x)/x|` = − log|x| + log |c|

∴ `1/2 log|y^2| - 1/2 log|(2y + x)/x|` = log |c|

∴ `1/2 log|y^2/((2y + x)/x)|` = log |c|

∴ `1/2 log|(xy^2)/(2y + x)|` = log |c|

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Chapter 1.8: Differential Equation and Applications - Q.5

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