solve 2sin(y^2)dx+xy cos(y^2)dy=0,y(2)=√π/2
Answers
Answer:
2sin(y
2
)dx+xycos(y
2
)dy=0
Using seperation of variable method,
xycos(y^2)cos(y
2
) dy=-2sin(y^2)dxdy=−2sin(y
2
)dx
\frac{ycos(y^2)}{sin(y^2)}dy=-\frac{2}{x}dx
sin(y
2
)
ycos(y
2
)
dy=−
x
2
dx
Integrating both the side \int ycot(y^2)dy=-\int \frac{2}{x}dx∫ycot(y
2
)dy=−∫
x
2
dx ....(1)
let y^2=t
2
=t
Diffrentiate with respect to x
2ydy=dt
ydy=\frac{dt}{2}
2
dt
Substitute y ^2
2
=t in equation 1
\int \frac{Cot tdt}{2}=-2logx+logc∫
2
Cottdt
=−2logx+logc
\frac{log|sin t|}{2}=logx^{-2}+logc
2
log∣sint∣
=logx
−2
+logc
log|sin (y^2)|^{0.5}=logx^{-2}+logclog∣sin(y
2
)∣
0.5
=logx
−2
+logc
log|sin (y^2)|^{0.5}=logx^{-2}clog∣sin(y
2
)∣
0.5
=logx
−2
c { using loga+logb=logab}
sin(y^2)^{0.5}=\frac{c}{x^2}sin(y
2
)
0.5
=
x
2
c
\sqrt{sin(y^2)}=\frac{c}{x^2}
sin(y
2
)
=
x
2
c
....(2)
at Y(2)=\sqrt{\frac{\Pi}{2}}
2
Π
\sqrt{sin(\frac{\Pi}{2})}=\frac{c}{2^2}
sin(
2
Π
)
=
2
2
c
1=\frac{c}{4}
4
c
c=4
Putting the value of c in equation 2
\sqrt{sin(y^2)}=\frac{4}{x^2}
sin(y
2
)
=
x
2
4
Step-by-step explanation:
this is the correct answer hope it helps you study well
