Solve differential equation (1+x^2)dy/dx+2xy = 4x^2

14/03/2025 by Team Gmath

Question 2:- Solve the following differential equation :

$(1+x^2)\dfrac{dy}{dx} + 2xy = 4x^2$ .

Solve:- Given, $(1+x^2)\dfrac{dy}{dx} + 2xy = 4x^2$

⇒ $\dfrac{dy}{dx} + \dfrac{2x}{(1 + x^2)}y = \dfrac{4x^2}{(1+ x^2)}$ .

Compare, $\dfrac{dy}{dx} + PY = Q$

$P = \dfrac{2x}{1 + x^2}, Q = \dfrac{4x^2}{(1+ x^2)}$

$I.F. = e^{\int pdx}$

⇒ $I.F. = e^{\int \dfrac{2x}{1 + x^2} dx}$

Let, $1 + x^2 = t$

⇒ $2x dx = dt$

⇒ $dx = \dfrac{dt}{dx}$

⇒ $I.F. = e^{\int \dfrac{2x}{t} \dfrac{dt}{2x}}$

⇒ $I.F. = e^{\int \dfrac{1}{t} dt}$

⇒ $I.F. = e^{\log t}$

⇒ $I.F. = (1 +x^2)$

Solution, $y(I.F.) = \int Q (I.F.)$

⇒ $y.(1 + x^2) = \int \dfrac{4x^2}{(1+ x^2}.(1 + x^2)dx$

⇒ $y.(1 + x^2) = \int 4x^2dx$

⇒ $y. (1 + x^2) = \dfrac{4 x^3}{3} + C$

⇒ $y = \dfrac{4 x^3}{3(1 + x^2)} + \dfrac{C}{(1 + x^2)}$

Question 1:- Using integration, find the area of the region bounded by the line y = 5x + 2, the x- axis and the ordinates x = -2 and x = 2. [CBSE 2025]

Solution:- See full solution

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