Question

ANURAG UNIVERSITY T-B.Tech.
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9. The solution of the differential equation |(D)y = Q(x)is​

Answer 1

Step-by-step explanation:

Derive the solution of the linear differential equation: dydx+P(x)⋅y=Q(x)

Rewriting the given differential equation, we obtain: (Py−Q)dx+1⋅dy=0.

Let M=Py−Q,N=1. Then : ∂M∂y=My=P

and ∂N∂x=Nx=0.

Thus My−NxN=P(x). Thus, the integrating factor is I.F=e∫Pdx. Therefore e∫Pdx(Py−Q)dx+e∫Pdx⋅dy=0 is an exact differential equation.

The solution of this exact differential equation is ∫treat y as constantMdx+∫terms in N not containing x dy= constant

⟹∫e∫Pdx(Py−Q) dx+0=c

⟹y∫P e∫Pdx dx=∫e∫PdxQ dx+c.