# Solve: (x + y) dy = a2 dx - Mathematics and Statistics

Sum

Solve:

(x + y) dy = a2 dx

#### Solution

(x + y) dy = a2 dx

∴ dy/dx = a^2/(x+y) ...(i)

Put x + y = t  ...(ii)

∴ y = t - x

Differentiating w.r.t. x, we get

∴ dy/dx = dt /dx -1 ....(iii)

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

dt/dx -1 = a^2/t

∴ dt/dx = a^2/t + 1

∴ dt/dx = (a^2+t)/t

∴ t/(a^2+t)  dt = dx

Integrating on both sides, we get

int ((a^2+t) - a^2)/(a^2+ t)  dt = int dx

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

∴ t - a2 log |a2 + t| = x + c1

∴ x + y - a2 log |a2 + x + y| = x + c1

∴ y - a2 log |a2 + x + y| = c1

∴ y - c1 = a2 log |a2 + x + y|

∴ y/a^2 - c_1/a^2 = log |a^2 + x + y|

∴ a^2 + x + y = e^(a^(y/2). e^(a^((-c1)/2)

∴ a^2 + x + y = ce^(a^(y/2)  … [ c =e^(a^((-c1)/2)]]

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
Miscellaneous Exercise 8 | Q 4.07 | Page 173