If the solution of the differential equation dydx−y(x4+3x2(x2+1)2)=(4x+3)ex3x2+1 is in the form y=f(x) (where f(0) =1), the value of f(1) + f(–1) is equal to ke, then k is equal to _____

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

If the solution of the differential equation dydx−y(x4+3x2(x2+1)2)=(4x+3)ex3x2+1 is in the form y=f(x) (where f(0) =1), the value of f(1) + f(–1) is equal to ke, then k is equal to _____

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

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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), the value of f(1) + f(–1) is equal to $k \sqrt{e}$ , then k is
equal to _____

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answer is D.

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Detailed Solution

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