Find the equation of curve whose slope at any point is sinx and which passes through the point (π,1).
Answers
Given slope of curve at any point =y+2x
i.e.
dx
dy
=y+2x
⇒
dx
dy
−y=2x ....(1)
It is of the form
dx
dy
+Py=Q
Here, P=−1,Q=2x
I.F.=e
∫pdx
=e
∫−1dx
=e
−x
∴ The solution of eqn (1) is given by
ye
−x
=∫2xe
−x
dx+c
=2[x(−e
−x
)−∫1.(−e
−x
)dx]+c
=2[−xe
−x
−e
−x
]+c
⇒y=−2x−2+ce
x
which is the equation of the family of curves.
Since, it passes through the origin (0,0),
0=−2+c or c=2
∴ The required equation of the curve is 2x+y+2=2e
x
Answer:
Answer
Given slope of curve at any point =y+2x
i.e.
dx
dy
=y+2x
⇒
dx
dy
−y=2x ....(1)
It is of the form
dx
dy
+Py=Q
Here, P=−1,Q=2x
I.F.=e
∫pdx
=e
∫−1dx
=e
−x
∴ The solution of eqn (1) is given by
ye
−x
=∫2xe
−x
dx+c
=2[x(−e
−x
)−∫1.(−e
−x
)dx]+c
=2[−xe
−x
−e
−x
]+c
⇒y=−2x−2+ce
x
which is the equation of the family of curves.
Since, it passes through the origin (0,0),
0=−2+c or c=2
∴ The required equation of the curve is 2x+y+2=2e
x