# Solve the following differential equation. xy dydx=x2+2y2 - Mathematics and Statistics

Sum

Solve the following differential equation.

xy  dy/dx = x^2 + 2y^2

#### Solution

xy  dy/dx = x^2 + 2y^2

∴ dy/dx = (x^2 + 2y^2)/(xy) …(i)

Put y = tx  ...(ii)

Differentiating w.r.t. x, we get

dy/dx = t + x dt/dx  ...(iii)

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

t +x dt/dx = (x^2 + 2t^2 x^2)/(x(tx))

∴t +x dt/dx = (x^2 (1 + 2t^2))/(x^2t)

∴ x dt/dx = (1 + 2t^2)/t  - t = (1+t^2)/t

∴ t / (1+t^2) dt = 1/xdx

Integrating on both sides, we get

1/2 int (2t)/(1+t^2) dt = int dx/x

∴ 1/2 log |1+ t^2| = log|x| + log |c_1|

∴ log |1 + t2 | = 2 log |x| + 2log |c1|

= log |x2| + log |c|    …[logc12 = log c]

∴ log |1 + t2| = log |cx 2|

∴ 1 + t2 = cx2

∴ 1+ y^2/x^2 = cx^2

∴ x2 + y2 = cx4

Concept: Differential Equations
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Differential Equation and Applications
Exercise 8.4 | Q 1.6 | Page 167