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A linear differential equation is of the form

dy/dx + Py = Q ..... (1) Where P and Q are constants or functions of x.

First multiply both its sides by a function h(x).

h(x) dy/dx + Pyh(x) = Qh(x) ..... (2)

The function h (x), such that the right hand side is a derivative of the product of y and h(x).

h(x) dy/dx + Pyh(x) = d/dx[yh(x)]

â‡’ h(x) dy/dx + Pyh(x) = h(x) dy/dx + yh'(x)

â‡’ Pyh(x) = yh'(x)

â‡’ P h(x) = h'(x)

â‡’ P = h'(x)/h(x) (integrate on both the sides)

â‡’ âˆ« P dx = âˆ« h'(x)/h(x) dx

â‡’ âˆ« P dx = log(h(x))

â‡’ h(x) = e^{âˆ«Pdx}

Substitute the value of h(x) in equation 2.

â‡’ e^{âˆ«Pdx} dy/dx + Pe^{âˆ«Pdx} y = Q e^{âˆ«Pdx}

â‡’ d/dx (ye^{âˆ«Pdx}) = Qe^{âˆ«Pdx }(integrate both the sides of the equation)

â‡’ d(ye^{âˆ«Pdx}) = Qe^{âˆ«Pdx}dx

â‡’ âˆ« d(ye^{âˆ«Pdx}) = âˆ« Qe^{âˆ«Pdx}dx

â‡’ ye^{âˆ«Pdx} = âˆ« Qe^{âˆ«Pdx}dx

This is the required solution of the differential equation.

Integrating factor = e^{âˆ«Pdx}

The general solution of equations of type 1 can be obtained by the formula

y x (IF) = q x (I.F)dx

If x is replaced by y, and y is replaced by x in equation 1, then the linear differential equation

dx/dy + P(y)x = Q(y)

x(I.F) = âˆ« Q(y) x (IF)dy , where IF = e^{âˆ«Pdx}

A linear differential equation is of the form

dy/dx + Py = Q ..... (1) Where P and Q are constants or functions of x.

First multiply both its sides by a function h(x).

h(x) dy/dx + Pyh(x) = Qh(x) ..... (2)

The function h (x), such that the right hand side is a derivative of the product of y and h(x).

h(x) dy/dx + Pyh(x) = d/dx[yh(x)]

â‡’ h(x) dy/dx + Pyh(x) = h(x) dy/dx + yh'(x)

â‡’ Pyh(x) = yh'(x)

â‡’ P h(x) = h'(x)

â‡’ P = h'(x)/h(x) (integrate on both the sides)

â‡’ âˆ« P dx = âˆ« h'(x)/h(x) dx

â‡’ âˆ« P dx = log(h(x))

â‡’ h(x) = e^{âˆ«Pdx}

Substitute the value of h(x) in equation 2.

â‡’ e^{âˆ«Pdx} dy/dx + Pe^{âˆ«Pdx} y = Q e^{âˆ«Pdx}

â‡’ d/dx (ye^{âˆ«Pdx}) = Qe^{âˆ«Pdx }(integrate both the sides of the equation)

â‡’ d(ye^{âˆ«Pdx}) = Qe^{âˆ«Pdx}dx

â‡’ âˆ« d(ye^{âˆ«Pdx}) = âˆ« Qe^{âˆ«Pdx}dx

â‡’ ye^{âˆ«Pdx} = âˆ« Qe^{âˆ«Pdx}dx

This is the required solution of the differential equation.

Integrating factor = e^{âˆ«Pdx}

The general solution of equations of type 1 can be obtained by the formula

y x (IF) = q x (I.F)dx

If x is replaced by y, and y is replaced by x in equation 1, then the linear differential equation

dx/dy + P(y)x = Q(y)

x(I.F) = âˆ« Q(y) x (IF)dy , where IF = e^{âˆ«Pdx}