Solve the differential equation dy - dx -3 y x -sin 2 x - y -2 when x -2-

Dy/dx + ( 3 cot x)y = sin 2x It is an L.D.E in y P = 3 cot x, Q = sin 2x I.F = e^∫ pdx = e^ 3logsinx = 1/sin^3x G.S: y (IF) = ∫ Q (IF) dx + c y/sin^3x = ∫ 2si ...

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Standard XII

Mathematics

Differential Equations Definition

Solve the dif...

Question

Solve the differential equation $\frac{d y}{d x} - 3 y c o t x = s i n 2 x, y = 2 when x = \frac{π}{2}$ .

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Solution

$\frac{d y}{d x}$ + (-3 cot x)y = sin 2x

It is an L.D.E in y

P = -3 cot x, Q = sin 2x

I.F = $e^{\int p d x}$ = $e^{- 3 l o g s i n x} = \frac{1}{s i n^{3} x}$

G.S: y (IF) = $\int Q$ (IF) dx + c

$\frac{y}{s i n^{3} x}$ = $\int$ $\frac{2 s i n x c o s x}{s i n^{3} x}$ dx + c

y $c o s e c^{3} x$ = 2 $(\frac{- 1}{s i n x}) + c$ .....(1)

(1) passes through $(\frac{π}{2}, 2)$

$\Rightarrow$ 2 = - 2 + c $\Rightarrow$ c = 4

$\Rightarrow$ y $c o s e c^{3}$ x = $\frac{- 2}{s i n x}$ + 4

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Solve the differential equation dydx − 3y cotx = sin2x, y = 2 when x =π2 .

dydx + (-3 cot x)y = sin 2x It is an L.D.E in y P = -3 cot x, Q = sin 2x I.F = e∫pdx = e−3logsinx=1sin3x G.S: y (IF) = ∫Q (IF) dx + c ysin3x = ∫ 2sinx cosxsin3x dx + c y cosec3x = 2 (−1sinx)+c .....(1) (1) passes through (π2,2) ⇒ 2 = - 2 + c ⇒ c = 4 ⇒ y cosec3 x = −2sinx + 4

Solve the differential equation $\frac{d y}{d x} - 3 y c o t x = s i n 2 x, y = 2 when x = \frac{π}{2}$ .