Assume x,y,t > 0. Solve the differential equation:

Question

t*(dx/dt) = (1+2ln*t)tanx?

Answer 1

Given equation

t (dx/dt) = (1 + 2lnt)tanx

Step1 : First separate the variables

t (dx/dt) = (1 + 2lnt)tanx

seperate t and x terms

(dx) / tan x = (1 + 2lnt) (dt / t)

( 1 / tan x ) dx = [ ( 1/t ) + ( 2lnt ) / t ] dt

Step2 : Apply integration on both sides

∫ ( 1 / tan x ) dx =  ∫ [ ( 1/t ) + ( 2lnt ) / t ] dt

∫ ( 1 / tan x ) dx =  ∫ ( 1/t ) dt + ∫ [( 2lnt ) / t ] dt       ---------------- (1)

Solve three integral parts seperately in equation (1)

Part - 1 : ∫ ( 1 / tanx ) dx

= ∫ ( cotx ) dx

= ln(sinx)

Part - 2 : ∫ ( 1/t ) dt

= ln(t)

Part - 3 : ∫ [( 2lnt ) / t ] dt

= 2 ∫ [(lnt) / t ] dt

= 2 ∫ [(lnt)(dt/t)]

Using u-substitution method

u = lnt

du = (1/t) dt

2 ∫ udu

2 [ u²/2 ] + c

Substitute u = ln(t)

(ln(t))²

From equation (1)

∫ ( 1 / tan x ) dx =  ∫ ( 1/t ) dt + ∫ [( 2lnt ) / t ] dt

ln(sinx) = ln(t) + (ln(t))² + c

ln(sinx) = ln(t) + (ln(t))² + c