COMEDK 2024 Shift 2 Paper

To determine the general solution of the given differential equation:
x‌

dy

dx

=y+x‌tan(‌

y

x

)
we can use a substitution method. Let's set:
v=‌

y

x

‌‌
so that
y=vx
To find the derivative ‌

dy

dx

, we use the product rule:
‌

dy

dx

=v+x‌

dv

dx

Substitute this expression and y=vx into the original differential equation:
x(v+x‌

dv

dx

)=vx+x‌tan(v)
Simplify the equation:
xv+x2‌

dv

dx

=vx+x‌tan(v)
Cancel the xv term on both sides:
x2‌

dv

dx

=x‌tan(v)
Divide both sides by x :
x‌

dv

dx

=tan(v)

Separate variables in the differential equation:
‌

dv

tan(v)

=‌

dx

x

Integrate both sides:
∫cot(v)‌d‌v=∫‌

dx

x

The integral of cot(v) is:
ln|sin‌(v)|
and the integral of ‌

1

x

is:
ln|x|
So the equation becomes:

ln|sin‌(v)|=ln|x|+ln|C|
Rewrite this as:
ln|sin‌(v)|=ln|Cx|
which implies:
|sin‌(v)|=|Cx|
Since v=‌

y

x

, we get:
|sin‌(‌

y

x

)|=|Cx|
or simply:
sin‌(‌

y

x

)=Cx
where C is an arbitrary constant. Thus, the general solution of the differential equation is given by Option B:
sin‌(‌

y

x

)=Cx