Find the general solution of the differential equation d y d x + 2 y = sin x .

Given differential equation is d y d x + 2 y = sin x . . . . . . . . . . . . . . . . 1 Equation ( 1 ) is of the form d y d x + P y = Q Where, P = 2 and Q = sin x Therefore, integrating factor is IF = e ∫ P d x = e ∫ 2 d x = e 2 x ∫ 1 . d x = x General solution is y · IF = ∫ Q · ( IF ) d x + c ⇒ y · e 2 x = ∫ sin x · e 2 x d x + c … … . . . ( 2 ) I = ∫ sin x · e 2 x d x Let, u = sin x and v = e 2 x I = sin x · ∫ e 2 x d x - ∫ d d x ( sin x ) · ∫ e 2 x d x d x = sin x · e 2 x 2 - ∫ cos x · e 2 x 2 d x = sin x · e 2 x 2 - 1 2 ∫ cos x · e 2 x d x Again let u = cos x and v = e 2 x ⇒ I = sin x · e 2 x 2 - 1 2 cos x · ∫ e 2 x d x - ∫ d d x ( cos x ) · ∫ e 2 x d x d x ⇒ I = sin x · e 2 x 2 - 1 2 cos x · e 2 x 2 + 1 2 ∫ sin x · e 2 x d x ⇒ I = sin x · e 2 x 2 - 1 2 cos x · e 2 x 2 + I 2 ⇒ I + I 4 = sin x · e 2 x 2 - cos x · e 2 x 4 ⇒ 5 I 4 = sin x · e 2 x 2 - cos x · e 2 x 4 ⇒ I = e 2 x 5 ( 2 sin x - cos x ) Substituting I in 2 , we get y e 2 x = 1 5 ( 2 sin x - cos x ) e 2 x + c ⇒ y = 1 5 ( 2 sin x - cos x ) + c e - 2 x .

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 21 - Linear Differential Equations>EXERCISE>Q 16

HARD

12th CBSE

IMPORTANT

Earn 100

Find the value of $k$ , if the general solution of the differential equation $(x \log x) \frac{d y}{d x} + y = 2 \log x$ is $y \cdot (\log x) = (\log x)^{k} + c$ .

50% studentsanswered this correctly

Important Questions on Linear Differential Equations

HARD

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 21 - Linear Differential Equations>EXERCISE>Q 17

Find the general solution of the differential equation $x \frac{d y}{d x} + y = x \log x$ .

HARD

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 21 - Linear Differential Equations>EXERCISE>Q 18

Find the general solution of the differential equation $x \frac{d y}{d x} + 2 y = x^{2} \log x$ .

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 21 - Linear Differential Equations>EXERCISE>Q 19

Find the value of $k$ , if the general solution of the differential equation $(1 + x) \frac{d y}{d x} - y = e^{3 x} (1 + x)^{2}$ is $y = \frac{1}{k} (1 + x) e^{3 x} + c (1 + x)$ .

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 21 - Linear Differential Equations>EXERCISE>Q 20

Find $a$ , if the general solution of the differential equation $\frac{d y}{d x} + \frac{4 x}{(x^{2} + 1)} y + \frac{1}{{(x^{2} + 1)}^{2}} = 0$ is $y = \frac{- a}{(1 + x^{2})^{2}} + \frac{c}{(1 + x^{2})^{2}}$