Solve the following differential equation x2 dydx = x2 + xy − y2

प्रश्न

Solve the following differential equation

`x^2 ("d"y)/("d"x)` = x² + xy − y²

बेरीज

उत्तर

`x^2 ("d"y)/("d"x)` = x² + xy − y²

∴ `("d"y)/("d"x) = 1 + y/x - (y/x)^2` .....(i)

Put `y/x` = t .....(ii)

∴ y = tx

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)` = 1 + t − t²

∴ `x "dt"/("d"x)` = 1 − t²

∴ `"dt"/(1 - "t"^2) = ("d"x)/x`

Integrating on both sides, we get

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

∴ `1/2 log|(1 + t)/(1 - t)|` = log |x| + log |c₁|

∴ `log |(1 + y/x)/(1 - y/x)|` = 2log |x| + 2log |c₁|

∴ `log|(x + y)/(x - y)|` = log |x²| + log |c₁²|

∴ `log|(x + y)/(x - y)|` = log |c¹x²|

∴ `(x + y)/(x - y)` = c₁²x²

∴ `(x + y)/(x - y)` = cx², where c = c₁²

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Basic Concepts of Differential Equations

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Q 10 Q 9 Q 1

पाठ 1.8: Differential Equation and Applications - Q.5

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एससीईआरटी महाराष्ट्र Mathematics and Statistics (Commerce) [English] 12 Standard HSC

पाठ 1.8 Differential Equation and Applications
Q.5 | Q 10

एससीईआरटी महाराष्ट्र Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

पाठ 2.6 Differential Equations
Attempt the following questions III | Q 10

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