CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Differential Equation

9.6.Linear Differential Equations.

Q.1. Solve the following differential equation : sin x dy/dx + cos x.y = cos x.sin² x

Solution :

We have, sin x dy/dx + cos x.y = cos x.sin² x
Or, dy/dx + cot x = cos x.sin x
This is a linear differential equation [dy/dx + P.y = Q]
Here P = cot x, Q = sin x.cos x
I.F. = e∫Pdx = e∫cot xdx = elog sin x = sin x
Therefore, the required solution of differential equation is
y. sin x = ∫cos x. sin x . sin x dx + c
= ∫ sin² x cos x dx + c [Put sin x = t , then cos x dx = dt]
= ∫t² dt + c
= t³/3 + c
= sin³ x/3 + c
Or, y = (1/3) sin² x + c cosec x. [Ans.]

Q.2. Solve the following differential equation : dy/dx – y/x = 2x².

Solution :

We have, dy/dx – (1/x).y = 2x²
This is a linear equation where, P = –1/x and Q = 2x².
Therefore, I.F. = e∫P.dx = e∫–1/x.dx = e–log x = e log x –1 = x –1.
Hence, solution of the given differential equation is y.x –1 = ∫2x².x –1 dx + c = 2∫x dx + c = 2.x²/2 + c = x² + c.
Thus y = x³ + cx. [Ans.]

Q.3. Solve the following differential equation : dy/dx + (sec x).y = tan x.

Solution :

We have, dy/dx + (sec x) .y = tan x ------------------- (1)
This is a linear differential equation where, P = secx and Q = tan x.
Now I.F. = e^∫Pdx
= e^{∫sec x dx}
= e^{log(sec x + tan x)}
= sec x + tan x.
Therefore, the solution of differential equation is :
y.(sec x + tan x) = ∫tan x (sec x + tan x)dx + c.
= ∫(sec x tan x + tan² x)dx + c.
= ∫sec x tan x dx + ∫sec² x dx – ∫1.dx + c.
= sec x + tan x – x + c.
Or, y = sec x + tan x – x + c. [Ans.]

Q.4. Solve the following differential equation : dy/dx + 2tan x .y = sin x.

Solution :

We have, dy/dx + 2tan x .y = sin x ----------------- (1)
This is a linear differential equation where, P = 2tan x and Q = sin x.
I.F. = e^∫Pdx = e^{∫2tan x dx}
= e^{2logsec x}
= e^logsec2 x
= sec2 x.
Therefore, solution of (1) is given by
y.sec2 x = ∫sin x.sec2 xdx + c
= ∫tan x.sec xdx + c
= sec x + c
Therefore, y = cos x + c cos2 x.

Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289