The eigenfunctions for y"- 2y'+ Xy = 0, y(0) = 1, y(1) = 0 are o {c² sin(nrx)} 1 00 O {e sin(")}n-1 2 o {e- sin(nrx)} 1 O {e? sin(") } O {e" sin(nnx)}n 1 o {e * sin(") 2 n=1 O {e-2 sin(naI)}n=1 00 O {e 2" sin(")}* n=1

Q: x" dx = x2 +1 ( x"-2(x2 +1 – 1) x² + 1 67. (а) 1, dx = x*-2 dx = п — 1 - In-2 - x2 + 1 (b) lo =…

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Q: 7. Given y =x*in*, calculate y'. sinx sinx A) y'=xin* (sin.x Inx + cos x Inx) C) y'=x*inx + cos x…

A: For the solution of the problem follow the next step.

Q: = In (2x) + cos (6x)

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Q: d If g(x)=- -S*[e' - In (21+1)]dt, find g(x) Se dx 4 A (2x+1)² 4 B e+ (2x + 1)² Ⓒe* -In (2x+1) 2 D…

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Q: y'''+y''=3 (Non Homogenous DE)

A: y'''+y''=3 (Non Homogenous DE)

Q: xyz" + (y +z)' + x'yz z(xy' + z)(x + y')

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Q: Find y'' if y = 6x sin x. A. y' = 6 cos x - 12x sinx B. y'= - 12 cos x + 6x sin x C. y' = 12 cos x -…

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Q: 16. (D' + D² + D+ 1)y = e* +e* + sinx %3D

A: We have to find the general solution of the differential equation D3+D2+D+1y=ex+e-x+sinx .

Q: Which one of these equations is not separable? (1) xy' = y ln |x| (2) y' = In y' (3) = x + sin(x)…

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Q: Show that \{e ^ (- x), x * e ^ (- x)\} is a fundamental set of solutions of y^ prime prime +2y^…

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Q: 9) 2y' = y cot(x) 10) y'+ csc(y) =0 11) y' = (1+x)(l+y³) 12) yy' = 0.5 sin (@x)

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Q: Find y' in y=sin¹ (5x5+2x) + Tan-¹ (x²) 1 1 1+x² -(5x5+2x)² A. y' B. y' C. y' D. y' = = 25x¹+2…

A: Given:y=sin-1(5x5+2x)+tan-1(x2)       y'=dydx=dsin-1(5x5+2x)+tan-1(x2)dx…

Q: Q/ ind dy if 24-sini' (xe!)=sin'(x*)+ cos (x")

A: Given that      2y - sin-1(x2+1) = sin2(x4) + cos2(x4) Here we have to find dy/dx.

Q: Then y' =

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Q: 21. If y=(x* +sin x)", then dy dx (A) 6(x*+sinx) (4x-cos.x) (B) 6(x*+sinx) (4x' +cosx) (C)…

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Q: bn =1/n f(x) cos nx dx bn" f(x) cos nx dx bn fo) sin nx dx bn -1/m f(x) sin nx dx

A: Last option is correct.

Q: 3) = + sin (4) r = y(In y- In r+1)

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Q: #23) y'(x) = 4 tan 2x-3 los x Saj (x) dx = S 4 tom 2x - 3 los x dx 3(x) = 4 S sin 2x dx - S3 losx.…

A: Here your answer is,-2lncos2x-3sinx+Cand the answer given in the book is,2lnsec2x-3sinx+C Here both…

Q: 3) y' + y = cos(e*)

A: We have to find the general solution of the first order linear differential equation

Q: 2) Show that La- (1 – x²)P',(x)P'm(x) dx : 21(1 + 1) Sim. 21 +1

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Q: 4) y'-4y = 2x-4x 6) y'+2y = cos(x) 8) y'=(y-1)cot(x)

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Q: Find dy/dx for y = 3l(sin-(e-3**)12 dy 3(sin"(e-*)* In3 2(sin-"(e-3x*)- dx 1(-6х) V1-e-6x2

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Q: (cas(r) – 1) = -sin(x)(?-y),>( ®* ) = 0

A: the solution of given initial value problem is:

Q: (x) = [*cos(t¹) dt, then g'(x) cos(x7) — sin(x7) — sin(77) + sin(x²) cos(77) - cos(x7)

A: Given query is to find the derivative.

Q: Find f'(x). f(x) = = O V 10x - sin²(7x) 1 f'(x) = = 2√/10 - 14cos(7x) 1 f(x) 2√10x - sin² (7x) 10…

A: This question is related to differentiation, We will solve it using given information.

Q: y"'+4y"+9y'+10y=e^-d(sind)

A: Explanation of the solution is given below....

Q: Solve for DE y" + 4y' + 3y = cos(e^x)

A: General solution of differential equation is sum of complementary solution and particular solution.

Q: If f(x) = e2" [sin(=)]°, what is f'(x)?

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Q: d[cos(x*)] dx (2k+1)! (-1)*2,* - (2k)! .(-1)시2x4~2 (2k+1)! (-1)*2x*-3 E (2k+1)! None (-1)*-'2x*-1 E…

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Q: 5. (a) C Evaluate sinxcos³ xdx 1 1 sin³ x+-sin³ x+C 3 5 3 sin x sin³ x+C 1 (b) cos³x-cos³x + 1 (d) 3…

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Q: 2) x³y? = 2x + 2y3 3) y = sin?(x) – cos³(2x) + 2 www. .

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Q: (2xe" + e*)dx + (x² + 1)e'dy = 0

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