BITSAT 2024 Shift 1 Paper

To solve the given differential equation
(x+1)‌

dy

dx

−y=e3x(x+1)2
we start by rewriting it in the standard linear form:
‌

dy

dx

−‌

y

x+1

=e3x(x+1)
Here, it's a first-order linear differential equation of the form
‌

dy

dx

+P(x)y=Q(x)
where P(x)=−‌

1

x+1

and Q(x)=e3x(x+1).
Step 1: Calculate the Integrating Factor (IF)
The integrating factor is given by:
IF=e∫P(x)‌dx=e∫−‌

1

x+1

‌dx=e−ln(x+1)=‌

1

x+1

Step 2: Determine the Solution
Using the integrating factor, the solution of the differential equation is:
y⋅IF=∫Q(x)⋅IF‌dx
Substituting the known values:
y⋅‌

1

x+1

=∫e3x(x+1)⋅‌

1

x+1

‌dx
This simplifies to:
‌

y

x+1

=‌

e3x

3

+C′
Therefore, the complete solution is:

‌

3y

x+1

=e3x+C′
where C′=C∕3 representing an integration constant.