Find the differential equation whose general solution is given by y = C1e4x + c2ex + x2
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Class 12
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>>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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Verified by Toppr
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