Question

if ay=sin(x+y) prove that y(1+(dy/dx))^3=0​

Answer 1

it has given that ay = sin(x + y)

we have to prove that d²y/dx² + y[1 + (dy/dx)]³ = 0

proof : equation is ... ay = sin(x + y)

differentiating with respect to x,

a dy/dx = cos(x + y) [1 + dy/dx] ....(1)

again differentiating with respect to x,

a d²y/dx² = -sin(x + y) (1 + dy/dx)² + cos(x + y) d²y/dx²

⇒a d²y/dx² = - ay(1 + dy/dx)² + cos(x + y)d²y/dx²

⇒a d²y/dx² = -ay(1 + dy/dx)² + ady/dx/(1 + dy/dx) d²y/dx² [ from eq (1) ]

⇒a d²y/dx² = [-ay(1 + dy/dx)³ + a dy/dx d²y/dx²]/(1 + dy/dx)

⇒d²y/dx²(1 + dy/dx) + y(1 + dy/dx)³ - dy/dx d²y/dx² = 0

⇒d²y/dx²(1 + dy/dx - dy/dx) + y(1 + dy/dx)³ = 0

⇒d²y/dx² + y(1 + dy/dx)³ = 0

hence proved.

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