Question

Problem 3. For the equations below, find an appropriate integrating factor and solve the initial value problems.

Answer 1

If a differential equation of the form M(x, y)dx + N(x, y)dy = 0 is not exact, i.e., partial differential M/ partial differential y(x, y) notequalto partial differential N/ partial differential x(x, y) Then we can find an integrating factor mu(y) = e^integral fmu(y)dy where mu(y) = partial differential m/ partial differential y - partial differential N/ partial differential x/ - mu(x, y). This is only when the RHS is dependent only on y We have partial differential M/ partial differential y = 0, partial differential N/ partial differential x = 2xy since partial differential M/ partial differential y notequalto partial differential N/ partial differential x the equation is not exact consider mu(y) = partial differential M/ partial differential y - partial differential N/ partial differential x/-M(x, y) = 0 - 2xy/-x = -2xy/-x = 2y left arrow dependent only on y Thus we have the integrating factor mu(y) = e^integral mu(y)dy = e^integral