. The general solution of the differential equation

Best Answer

Explanation:

We know that if the equation is dy/dx+P(x)y=Q(x) then the integrating factor =e^∫P(x)dx

Given the differential equation is 

y'+y ϕ'(x)- ϕ(x). ϕ'(x)=0

⇒y'+ ϕ'(x)y= ϕ(x). ϕ'(x)

Then integrating factor =e^∫ϕ'(x)dx=e^ϕ(x)

Multiplying both sides by integrating factor we get, 

e ^ϕ(x)y'+e ^ϕ(x)' ϕ(x)y=e ^ϕ(x)(x) ϕ'(x)

⇒d(y.e ^ϕ(x))=e ^ϕ(x)(x) ϕ'(x)

Integrating both sides,

∫d(y.e ^ϕ(x))=∫e ^ϕ(x) ϕ(x) ϕ'(x)dx

⇒ye ^ϕ(x)=ϕ(x)∫e^ϕ(x)∫ϕ'(x)dx-∫(d( ϕ(x))/dx ×∫e^ϕ(x) ϕ'(x)dx)dx

⇒ye ^ϕ(x)= ϕ(x)e^ϕ(x)-∫ϕ'(x)e ^ϕ(x)dx

⇒ye ^ϕ(x)= ϕ(x)e ^ϕ(x)-e ^ϕ(x)+c, where c is an arbitrary constant 

⇒y= ϕ(x)-1+ce^- ϕ(x), multiplying both sides by e^- ϕ(x)

Final Answer:

Hence the correct option is (A). i.e y=ce ^-ϕ(x)+ ϕ(x)-1