)2+16 This is a homogenous different equation. Let
y
x
=v ⇒y=vx ⇒
dy
dx
=v+x
dv
dx
∴v+x
dv
dx
=v+√v2+16 ⇒x
dv
dx
=√v2+16 ⇒
dv
√v2+16
=
dx
x
Integrating both sides, we get ∫
dv
√v2+16
=∫
dx
x
⇒ln|v+√v2+16
|=lnx+lnc ⇒v+√v2+16=cx Now putting, v=
y
x
, we get
y
x
+√
y2
x2
+16=cx ⇒
y
x
+√
y2+16x2
x2
=cx ⇒y+√y2+16x2=cx2...... Given, y(1)=3 ∴ When x=1 then y=3. Putting in equation (1) we get, 3+√9+16=c.1 ⇒c=8 ∴ Solution of equation, y+√y2+16x2=8x2 Now, y(2) means when x=2 then y= ?