Math, asked by krity341, 1 day ago

Find the differential equation whose general solution is given by y = C1e4x + c2ex + x2 ​

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Answered by knigam260
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Class 12

>>Maths

>>Differential Equations

>>Formation of Differential Equation

>>The differential equation whose general

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The differential equation whose general solution is given by, y=(c

1

cos(x+c

2

)−(c

3

e

(−x+c4)

)+(c

5

sinx), where c

1

,c

2

,c

3

,c

4

,c

5

, are arbitrry constants, is

Medium

Solution

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Correct option is B)

y=c

1

cos(x+c

2

)−(c

3

e

−x+c

4

)+(c

5

sinx)

⟹y=c

1

(cosxcosc

2

−sinxsinc

2

)−(c

3

e

c

4

)+(c

5

sinx)

⟹y=(c

1

cosc

2

)cosx−(c

1

sinc

2

−c

5

)sinx−(c

3

e

c

4

)e

−x

⟹y=lcosx+msinx−ne

−x

; where l,m,n are arbitrary constant ...(1)

dx

dy

=−lsinx+mcosx+ne

−x

...(2)

dx

2

d

2

y

=−lcosx−msinx−ne

−x

...(3)

dx

3

d

3

y

=lsinx−mcosx+ne

−x

...(4)

From equation (1) + (3),

dx

2

d

2

y

+y=−2ne

−x

...(5)

From equation (2) + (4),

dx

3

d

3

y

+

dx

dy

=2ne

−x

...(6)

From equation (5) + (6),

dx

3

d

3

y

+

dx

2

d

2

y

+

dx

dy

+y=0

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