If the solution of the differential equation dydx−yx4+3x2x2+12=(4x+3)ex3x2+1 is in the form y=f(x) (where f(0)=1)then the value of f1+f-1 is equal to _____

IF=e−∫x4+3x2x2+12dx=e−x3x2+1⇒ye−x3x2+1=∫(4x+3)dx⇒f(x)=y=2x2+3x+1ex3x2+1⇒f(1)=6e&f(−1)=0⇒f(1)+f(−1)=6e

If the solution of the differential equation dydx−yx4+3x2x2+12=(4x+3)ex3x2+1 is in the form y=f(x) (where f(0)=1)then the value of f1+f-1 is equal to _____

IF=e−∫x4+3x2x2+12dx=e−x3x2+1⇒ye−x3x2+1=∫(4x+3)dx⇒f(x)=y=2x2+3x+1ex3x2+1⇒f(1)=6e&f(−1)=0⇒f(1)+f(−1)=6e

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If the solution of the differential equation $\frac{d y}{d x} - y (\frac{x^{4} + 3 x^{2}}{{(x^{2} + 1)}^{2}}) = (4 x + 3) e^{\frac{x^{3}}{x^{2} + 1}}$ is in the form $y = f (x) (where f (0) = 1)$ then the value of $f (1) + f (- 1)$ is equal to _____

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$2 \sqrt{e}$

$3 \sqrt{e}$

$4 \sqrt{e}$

$6 \sqrt{e}$

answer is D.

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Detailed Solution

$\begin{aligned} I F = e^{- \int \frac{x^{4} + 3 x^{2}}{{(x^{2} + 1)}^{2}} d x} = e^{\frac{- x^{3}}{x^{2} + 1}} \\ \Rightarrow {ye}^{\frac{- x^{3}}{x^{2} + 1}} = \int (4 x + 3) dx \\ \Rightarrow f (x) = y = (2 x^{2} + 3 x + 1) e^{\frac{x^{3}}{x^{2} + 1}} \\ \Rightarrow f (1) = 6 \sqrt{e} & f (- 1) = 0 \\ \Rightarrow f (1) + f (- 1) = 6 \sqrt{e} \end{aligned}$