Consider the ODE dy arctanh(x) 2 da x² ey (0 < x < 1) x (a) Make the substitution u = =e" and show that the ODE reduces to (b) Solve the ODE for du da' du da arctanh(x) 2u 2:² x and hence find y(x).

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Q: Find f. f"(x) = 5e* – 8 sin(x), f(0) = 3, = 0 %3D %3D f(x) = IN

A: Answer and explanation is given below

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Q: Evaluate: 4S d sin(t?)dt dx x 3x²sin(x“) – sin(x²) b. 2x°cos(x°) – 2xcos(x²) a. 3x²cos(x^) – cos(x²)…

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A: Solution of given question is followed by:

Q: If f(x) = cos x, then f'() = A. B. - C. - C. D. √3

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Q: F(s) = Le 4t cos(5t)}

A: We have to find  ℒe-4tcos5t Remember : ℒeatf(t)=Fs-a where  ℒf(t)=Fs

Q: Obtain general solution of ODE y''-y=exp(x) + cos(2x)

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A: Consider the function  f(x)=3cos3ln(x)

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Q: (a) [In(4) - 2e¹³] dx d b) dx d (c) [2²² - 1 (2²)] = 4e³x In(x [sec (12r) + tan-¹(2x)]

A: NOTE: Refresh your page if you can't see any equations. .

Q: Show that d 1 In cos(x) os(x) - csc(x) dx 1+ cos(x

A: We have to proof

Q: (b) y" - y = cosh x .

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