Question

Find general and particular solution of the following equation
dy/dx +10y=15; y(0)=0

Answer 1

Solution:

The given diffentalial equation is

    dy/dx + 10y = 15 ..... (1)

Comparing (1) with general form of linear equation

    dy/dx + Py = Q (P, Q are either functions of x or constants), we get

    P = 10 and Q = 15

∴ integrating factor (I.F.) = $e^{\int P\:dx}$

    = $e^{\int 10dx}$

    = $e^{10x}$

Multiplying (1) by I.F. = $e^{10x}$ , we get

    $\frac{d}{dx}(ye^{10x})=15e^{10x}$

or, $d(ye^{10x})=15e^{10x}\:dx$

On integration, we get

    $ye^{10x}=15\int e^{10x}\:dx+C$

where C is constant of integration

or, $ye^{10x}=\frac{15}{10}e^{10x}+C$

or, $ye^{10x}=\frac{3}{2}e^{10x}+C$

or, $y=\frac{3}{2}+Ce^{-10x}$

∴ the general solution is

    $y=\frac{3}{2}+Ce^{-10x}$

Given y(0) = 0, i.e., when x = 0, y = 0

Then, 0 = 3/2 + C

   i.e., C = - 3/2

Hence, the particular solutions is

    $y=\frac{3}{2}-\frac{3}{2}e^{-10x}$

i.e., $y=\frac{3}{2}(1-e^{-10x})$