x dx 1. Complete the square to get J 21-(x+3)² and then use the substitution x + 3 = V21 sin u so that dx = V21 cos u du.

Q: ZX = 24, XY = 10, and YZ = 26. Find cos Y. %3D

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Q: y"+4y' + 3y = sin e*

A: Solving the differential equation by variation of parameters method

Q: Q.1B) Solve the follwing diferential equation: 1 dy x dx siny + ycosy 2logx +1

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Q: Q) Lind d/dx of ® xe 1o x+3J = 0 2 2sinx cos y =1 ® x -se cs

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Q: H.wl: Find L{e-3t St. sin 2t.dt p1 p2

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Q: Q4) w(x,y)=x siny +x²- (2x²+3y)² find dw

A: We have to find the total derivative

Q: Solve for C using the form f(x,y)=C. Given [2x + ycos(xy)] dx + x cos(xy)dy = 0; f(1)=pi/6. 1…

A: We use the concept of exact differential equations to find the general solution to given…

Q: Q2 Find (d y/d x)& (dy/dx?) y 1/ cos xy ex =

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Q: Find the particular solution of 3 In x sec y dx – 5x sin? y dy = 0 given y(e³) = 7 %3D

A: Topic :- Differential equation

Q: Obtain the DE given the general solution . y = c, + C,x + C,x³ + (c, +c,x)e~Và« + (cq cos 2x + c,…

A: We will generate auxiliary equation from the roots of it

Q: Given z = 2 cos t + 35 and y= 9 sin? t da (i) dt dy (ii)- dt dy (iii) dx

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Q: (c) Given T" = 6x²y³ – cosx + 5y² – siny + 10x² – e-3x," determine (1) T' = S(T")ax T' = S(T")ay…

A: GivenT"=6x2y2-cos x+5y2-sin y+10x2-e-3xwe know…

Q: Solve the I.V.P using [x = zry]. y di dx + (y cos ( 1 ) - x) dy = 0, y (0) = 262

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Q: c. T(x) = x (**-cos(2x)). find Q at x = t/4. d. T(x) 100e-3x + In(4x), k=470, A= 0.01r and x= 1/3;…

A: c) Given, T(x)=x(x2-cos(2x)) at x=π2

Q: dz Find dy x? sin(2y – 5z) =1+y cos(6za)

A: The given equation is: x2sin(2y-5z)=1+ycos(6zx) To find ∂z∂y, differentiate above equation w.r.t. y…

Q: W: Solve heat eq. du at u (ort) = du (πT₁ t) = 0 ди dx and I. C₁ u(x₁0) = 3 sin (SX) ax² with b-Cs:

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Q: t (b) x(t) = S 3t cos(4t)dt

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Q: f(t) = e2t – 2t² cos 4t + 3t sin 6t – 2t + 5 %3D -

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Q: Find the general solution of (y + 2y) sin r²dr + V1 dy = dp. 0. %3D

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Q: Q2 Given z = xw? + cos y + wx , w= t , x = 2t2 , and y = 3t, find dt

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Q: Find dy and evaluate when x = T and dx 0.5 for the function y -10 cos (5x)

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Q: Given I = = 16 sint +6 and y = 16 cos t +2 da (i) dt 16* cos(t) dy (ii) 16*sin(t) %3D dt dy (iii)…

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Q: Q3.a) Find if y + SE rsecx -dt = xy sint+1 dz Q4.a) if z = sin²0, and 0 = w² + cos (w) find-

A: 3a):  y+∫xsecx11+sintdt=xydifferentiate both sides and use Leibniz rule for differentiating the…

Q: Solve the DE by eliminating the arbitrary constant of x sin y + (x^2)y = C

A: Given that the expression xsiny+x2y=C. Differentiate with respect to x.…

Q: 4. Find d²y/dx² for the following functions: a) y = sin x cos 2x %3D b) y = cos*(sec² 3x) c) y =…

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Q: Q2- : Classify and Solve; d?y + 9: dt dy cos? t + cosht + sinh t + sin? t +3 sin t - 3t %3D dt?

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Q: Find a positive value of k for which y=cos(kt) satisfies y''+25y=0.

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Q: Q7// find the general solution (sin-1x + 1 y)dx + sec-1xdy = 0 1 xVx² – |

A: The detailed solution is as follows below:

Q: FIND Y(S) OF THE GIVEN EQUATION. y" + 4y = cos(t-3) +4t, y(3)=0 y' (3)=7

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Q: (a) Find the particular solution of dy, 2y cOS I y(T) = 0 dx

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Q: Given x = sin3t, y = -cos3t, then find at t =. %3D dx а. 1 b. 3 с. О d. 2

A: We need to find dy/dx.

Q: Find the solution of dy + ycosx =0 dx O a. ye Sinx =0 O b. e Sinx = c O c. yeSinx = c O d. yeX = c

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Q: Solve the DE by Variation of Parameters (D³ + D° = 2D) y = 4e¯= cos z -2x COs I y (0) = 0.4, y' (0)…

A: The differential equation is D3+D2-2Dy=4e-2xcos x with the initial condition y0=0.4, y'0=-0.4,…

Q: f(t) = e3t + cos 6t – e3t cos 6t ; L{f(t)} =? %3D -

A: given ft=e3t+cos6t-e3tcos6t

Q: x dx V21-(x+3)² 1. Complete the square to get S and then use the substitution x + 3 = /21 sin u so…

A: we have the integral ∫xdx21-x+32 substituting the value x+3=21sin u we get   ∫21sinu.21cos…

Q: Determine whether y : y = x2 + 2 d²y sin x + x² is a solution to the DE dx2 %3D

A: We have to check whether  y = sin x + x2 is a solution to the DE  d2ydx2 + y = x2 + 2.

Q: Solve the equation (x^2-3y^2)dx + 2xydy = 0 using y=v.x and u transforms

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Q: T/2 %3D cos (2wt + O, + ©z)dt = 0 -/2

A: The solution are next step is

Q: Find the solution where y= 1 and dy/dx = 0 when x =0 and describe the behaviour of the solution for…

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Q: dx dt (b) 3x – 6x = sin 2t given that x = 1 when t = 0 (c) i- 7* + 12x = 2 given that at t = 0, x 1…

A: Since the questions contains many subparts as per the guidelines I am answering the first three…

Q: 4 T/2 a, = f(x)cos(n@,rkdr = "jxcos dx %3D sin 4 cos 2 2

A: Okay I am explaining the marked step.

Q: 3 T - 2 Evaluate: J. Soves y cosx³dy dx

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Q: Q2:B/ Find dy/dx f Via - "sin Vi dt y = + cos t dt

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Q: Q2 Find (d y/d x)& (dy/dx?) y 1/ cos xy ex %3D

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Q: Q8// find the general solution (tan-'x + (cotx)y)dx + (lnsinx)dy = 0

A: The differential equation is given as (tan-1x+(cotx)y)dx+(ln sinx)dy=0 Compare it with Mdx+Ndy=0…

Q: J1+ sin(wt) – cos(wt) 0<t<T sin(wt) – 2 cos(wt) f(t) = -

A: The given problem is to find the Laplace transform of the given unit step function.

Q: Solve for C using the form f(x,y)=C. Given [2x + ycos(xy)] dx + x cos(xy)dy = 0; f(1)-pi/6. decimal…

A: differential equation

Q: * x(t) = t² - 2t aS, w = sin(etan(x*)x) cS lài dw dt

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Q: H.W- Compute y( E) with y-J = -8(Sinx+ cosX) 9(T)=0, ý(T) = -8 %3D

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Q: dy dx 2. 2X for y = cos (Inx + e) %3D

A: ddx(Cos(x)) = -sin(x)ddxex =exddx(log(x)) =1xddxxn=nxn-1Chain ruleddxf(f(x)) =f'(f'(x)) Here , In…