To solve the given differential equation
(x+1)‌−y=e3x(x+1)2we start by rewriting it in the standard linear form:
‌−‌=e3x(x+1)Here, it's a first-order linear differential equation of the form
‌+P(x)y=Q(x)where
P(x)=−‌ 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∫−‌‌dx=e−ln(x+1)=‌Step 2: Determine the Solution
Using the integrating factor, the solution of the differential equation is:
y⋅IF=∫Q(x)⋅IF‌dxSubstituting the known values:
y⋅‌=∫e3x(x+1)⋅‌‌dxThis simplifies to:
‌=‌+C′Therefore, the complete solution is:
‌=e3x+C′where
C′=C∕3 representing an integration constant.