Question

Solve the differential equation `dy/dx+2xy=x` by completing the following activity.

Solution: `dy/dx+2xy=x`       ...(1)

This is the linear differential equation of the form `dy/dx +Py =Q,"where"`

`P=square` and Q = x

∴ `I.F. = e^(intPdx)=square`

The solution of (1) is given by

`y.(I.F.)=intQ(I.F.)dx+c=intsquare  dx+c`

∴ `ye^(x^2) = square`

This is the general solution.

Answer 1

`dy/dx+2xy=x`       ...(1)

This is the linear differential equation of the form `dy/dx +Py =Q,"where"`

P = 2x and Q = x

∴ `I.F. = e^(intPdx)=e^(int2xdx) =e^(2^(x^2/2)) = bb(e^(x^2))`

The solution of (1) is given by

`y.(I.F.)=intQ(I.F.)dx+c

∴ `ye^(x^2) = intbb(x.e^(x^2))dx +c`      ...(2)

Put x² = t

∴ 2x dx = dt

∴ `x  dx = dt/2`

∴ (2) becomes

`ye^(x^2)=1/2 int e^t dt+ c=1/2 e^t + c`

`ye^(x^2) = bb(1/2 e^(x^2) + c)`

This is the general solution