Question

${4x}^{2} + 4x - 3$
​

Answer 1

GIVEN :–

• Diffrential equation –

$\\ \implies \bf(1 + x^2) \dfrac{dy}{dx} + 2xy = cosx$

TO FIND :–

• Solution of diffrential equation = ?

SOLUTION :–

$\\ \implies \bf(1 + x^2) \dfrac{dy}{dx} + 2xy = cosx$

$\\ \implies \bf\dfrac{dy}{dx} + \dfrac{2x}{(1 + x^2)}y = \dfrac{cosx}{(1 + x^2)}$

• Compare with –

$\\ \implies \bf\dfrac{dy}{dx} +Py = Q$

• So –

$\\ \implies \bf P =\dfrac{2x}{(1 + x^2)} \: \: and \: \: Q = \dfrac{cosx}{(1 + x^2)}$

• We know that –

$\\ \implies \bf I.F. = e^{\int P.dx}$

$\\ \implies \bf I.F. = e^{\int \frac{2x}{(1 + x^2)}dx}$

$\\ \implies \bf I.F. = e^{ \log{(1 + x^2)}}$

$\\ \implies \large{ \boxed{\bf I.F. = 1 + {x}^{2} }}$

• Solution :–

$\\ \implies \bf y(I.F.) = \int(I.F)Q.dx + c$

$\\ \implies \bf y(1 + {x}^{2} ) = \int\dfrac{cosx}{(1 + x^2)} \times (1 + {x}^{2}) .dx + c$

$\\ \implies \bf y(1 + {x}^{2} ) = \int\cos(x) .dx + c$

$\\ \implies \large { \boxed{\bf y(1 + {x}^{2} ) = \sin(x) + c}}$

okay

Answer 2

Step-by-step explanation:

$4 {x}^{2} + 4x - 3 \\ = 4 {x}^{2} - 2x + 6x - 3 \\ = 2x(2x - 1) + 3(2x - 1) \\ = (2x - 1)(2x + 3)$