if y=log(1+2t^2 + t^4) and x=tan^-1(t)
find d^2y/dx^2
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12th
Maths
Differential Equations
Solving Differential Equations - Variable Separable Method
If y = log ( 1 + 2t2+t4 ), ...
MATHS
If y=log(1+2t
2
+t
4
),x=tan
−1
t, find
dx
2
d
2
y
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ANSWER
y=log(1+2t
2
+t
4
)
Differentiating w.r.t. x, we get,
dx
dy
=
(1+2t
2
+t
4
)
1
(4t+4t
3
)
dx
dy
=
1+t
2
1
x=tan
−1
t
dx
dy
=
dt
dx
dr
dy
=
(1+t
2
)
1
(1+2t
2
+t
4
)
4t(1+t
2
)
=
1+2t
2
+t
4
4t(1+t
2
)
2
=
1+2t
2
+t
4
4t(1+t
4
+2t
2
=4t
dx
2
d
2
y
=d(
dx
dy
).
dx
dt
=4×(1+t
2
)
=4×(1+t
2
)
dx
2
d
2
y
=4(1+t
2
)
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