The given differential equation is,
e x dy+( y e x +2x )dx=0 dy dx = −( y e x +2x ) e x dy dx = −y e x e x − 2x e x dy dx +y=− 2x e x
The given differential equation is of the form dy dx +Py=Q.
Here,
P=1 Q=− 2x e x
Now, calculate the Integrating factor,
IF= e ∫ Pdx = e ∫ 1dx = e x
The solution of the differential equation is,
y( IF )= ∫ ( Q×IF )dx +c y e x = ∫ ( − 2x e x e x ) dx+c y e x =−2 ∫ x dx+c y e x =−2 x 2 2 +c
Further, simplify the equation,
y e x =− x 2 +c y e x + x 2 =c
Hence, the correct option is ©.