Find a particular solution satisfying the given condition for differential equations. dy\dx + 2xy = x , given that y = 0 when x =0

Question

Find a particular solution satisfying the given condition for \(\frac{dy}{dx}\) + 2xy = x , given that y = 0 when x =0 differential equations.

dy\dx + 2xy = x , given that y = 0 when x =0

Answer 1

Given Differential Equation :

\(\frac{dy}{dx}\) + 2xy = x

Formula :

i) \(\int\) xⁿ dx = \(\frac {x^{n+1}}{n+1}\)

ii) \(\int\) \((e^{kx})\) dx = \(\frac {e^{kx}}{k}\)

iii) General solution :

For the differential equation in the form of

\(\frac{dy}{dx} \, +Py\, =Q\)

General solution is given by,

y.(I.F.) = \(\int\) Q. (I.F.) dx + c

Where, integrating factor,

I.F. = \(e^{\int \,p\, dx}\)

Given differential equation is

\(\frac{dy}{dx}\) + 2xy = x …eq(1)

Equation (1) is of the form

\(\frac{dy}{dx} \, +Py\, =Q\)

where, P = 2x and Q = x

Therefore, integrating factor is

General solution is

Put, x²=t => 2x dx = dt

Substituting I in eq(2),

Therefore, general solution is

For particular solution put y=0 and x=0 in above equation,

Substituting c in general solution,

Multiplying above equation by \(\frac{2}{e^{x2}}\)

Therefore, particular solution is