1. (Theorem 2.1, Eq. 2.4) By employing change of variables r = ax +y and s=x, prove that the PDE: A Urr + B Ury +C Uyy = 0 has the following general solution: when: u(x, y) = F(2Ay – Bx) + xG(2Ay – Bx). A0, B²-4AC = 0

Q: C1 and C2 are arbitrary constants. Note: Y1(t) 2t and y2(t) = e2t are independent solutions of y" –…

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Q: The equations cos(u) + 10uy2 + e + 2 = 0 and 2ye" + 4x2v - 3xcos(z) +1 =0 are solved for u and v as…

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Q: xux − yuy + y²u = y²

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Q: Y4(t) = ¿et is a particular solution of y" – y = 2e3t. The general solution of /" – y = 2e3t is…

A: We find whether the given statements are true or false.

Q: d a value of r> 0 such that h(z) = z² +22³ + z + 1 has all its zeroes in D,(0).

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Q: 2. (Theorem 2.1, Eq. 2.5) By employing change of variables r = ax +y and s= bx +y, prove that the…

A: Using chain rule. Let u is a function of x and y,  r and s are both function of x and y then…

Q: 2. a) Find the general solution of (z²-2yz-y²)p+(xy+xz)q=xy-xz, b) which passes through the curve…

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Q: Let z = 8x2 + 7y2, and suppose that (x, y) changes from (2, -1) to (2.01, -0.98). (a) Compute Az. Az…

A: We have to find   dz and delta z and compare the the values.

Q: Let y1, Y2 be two independent solutions of (t – 1)y" – ty' + y = 0. If W (y1, Y2)(2) = e² , then…

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Q: (part 1 of 4) Determine fa- fy when 1. 2. 3. 4. 6. f(x, y) = 4x² + 4xy - 4y² + x + 2y. fx-fy = 12x -…

A: Topic = Derivative

Q: 11. Find the linear approximation to given function at indicated point (Баллов: 2) x2 z = 3 + +, Po…

A: Topic:- linear approximation

Q: 2) z(x, y) = (x² + 5y)³ (3y + 2x)

A: Let's find dz/dx and dz/dy.

Q: (a) Use the Intermediate Theorem to prove that Vx + Vx + 7 = 3 has a solution in the interval [0,…

A: We have to use intermediate value theorem.

Q: Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system y₁ =…

A: To Find : Which of the following function Vy1 , y2 is Lyapunov function for the dynamic system.…

Q: (D) e(-1-v2)x + cze(-1+v2)x If r2+2r-1 = 0 is the characteristic equation of a homogeneous D.E. with…

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Q: Let (x, Y,) be the solution of the following equations (2x)" = (3y)m3 3Inx = 2lny In 2 %3| %3D Then…

A: Introduction :

Q: The equations xcos(u) + 9uy2 + e² + 2 = 0 and 3ye" + 4x²v – 3æcos(2) +1 = 0 are solved for u and v…

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Q: 1. Verify that y = e²x + 3xe²x is a solution of the ODE y" - 4y + 4y = 0. (Show all work when…

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Q: The functions f1 (z), f2 (z) and f3 (z) defined in the following equations (being EC) are given. i.…

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Q: Į va | Vi(t? + 6t +7) dt. 4. Find

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Q: independent solutions of y''+ P(x)y'+ Q(x)y = 0. Assume that f1(x) has at

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Q: ) Find of the implicit function a? + 5xy + y? – 2x + y – 6 = 0 at the point 1. . (1, 1).

A: Topic:- derivatives

Q: (c) Consider the double integral || xy dA where D is bounded by the x-axis and the upper half…

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Q: If the functions y₁ and y2 are linearly independent solutions of y" + p(x)y +q(x)y = 0, then its…

A: Given the functions y1 and y2 are linearly independent solutions of y '" + p ( x ) y' + q ( x ) y =…

Q: If y, and y, are linearly independent solutions of ty" + 3y' + tey = 0 and if W(y,, y2)(1) = 6, find…

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Q: (2, 2) (0, 2) C1 C3 C2 (2, 0) 3 (0, 0) -1 1.The equation of C3 can be expressed as z= p+ qi +…

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Q: Verify that u(x, y) = f(r)g(y) is a solution of the PDE uury = uzuy for all pairs of (differen-…

A: Solution to the partial differential equation

Q: (14-17) find the solution of following derivative by Rung-Kutta method f(x,y)=-2x³ + 12x² - 20x +…

A: The given function is , .To Find: Solution of the derivative of above function using Runge-Kutta…

Q: (8) If w = 5x² + 2y², x= -3s+t, and y = s – 4t. Find and Əw as

A: The objective is to find the partial derivatives .

Q: d/dt ŷ = [3 1] ŷ, ý(0) = [8] [-2 1] | [6]

A: Given that dy^dt=31-21y^ with initial condition y^(0)=86 Let A=31-21 and eigenvalue of A is λ. Then…

Q: -2ux +4uy +4u = ex+³y.

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Q: If fx(x)= 1/(1+x²), and fy(y)= 1/(1+y"), X & Y are independent. Find fzw(z,w), if Z-6,5 XY and…

A: Given: fxx=11+x2 fy(y)=11+y2 Determine ∂(z,w)∂(x,y)…

Q: 3 4. If y = t3 + 3t? and x = t - 8t2, show that y" = 16t(t+2)(t-2)3

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Q: Let A(t) = 3ti+2]-4k, y B-21-21-3tk and C(t)=-1+3tj+6k. find Ax (B-C) dt

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Q: " – 36y" + 216y' = t² +6+t sint, - table form for Y(t) if the method of undete: be used. Do not…

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Q: (b) By determining constants c,, C2, C3 and c4, which are not all zero and are such that cifi(x) +…

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Q: The equations xcos(u) + luy² + e² + 2 = 0 and 9ye" + 4x²v – 3xcos(z) +1=0 are solved for u and v as…

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Q: 7. Find: t2 (a) / V1+ t3 (x2 + 2)(x – 3) dx (b) / * /. - 2.x (c) xV (16 – x4 dx

A: To evaluate.a ∫01 t21+t3 dtb ∫x2+2x-32x  dxc ∫-22 x16-x4 dx

Q: Solve for fazy for f(x, y, z) = 4x³y²z - e²zy¹ + + 4y - x¹6

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