EXERCISE 4 Find Vf(P) and D7f(P). (a) f(r, y, 2) = y² + z² – 4xz; U = } i – § j+ ; P(-2,1,3) %3D (b) f(r, y, 2) = 2r³ + ry? + xz²; Ū = }v21 j – V7 k; P(1,1, 1) %3D

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A: The given differential equation is y''+3y'+2y=12+ex. The objective is to solve the differential…

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Q: Transform the equation: 2x2y'' + xy' + 3y = 0 to the form of: a(d2y/dz2) + b(dy/dz) + cy = 0 (a,…

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Q: Solve for the orthogonal traysctories x*(3k-x) - ky². = x ? 2

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Q: the linearization of (fi,f2) = (x²y+ 3xy², 3x³ + 2xy) at (1,0) to approximate the value at…

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Q: Consider a function of 2 variable f(x, y) = 2x³y² – 5x²y* + 3x³ – 4y³ af and Əx af Find ду

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Q: The equations ry? + 5xzcos(u) + ye" = 10 and x³yz + xe" – 10u2v² = 30 are solved for u and v as…

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Q: Evaluate |(a,b)=(0,0) given that f(x, y) = 3x² + [y]; x = e²a+b₁ e²a+b; y = a² + t² + 1 да

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Q: Find Vƒ(P) and D7f(P). (a) f(r, y, 2) = yf + z² – 4xz; Ū = } i – $ j+ k; P(-2,1,3)

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Q: Find Duf at P. f(x, y, z) = 6x³y³z³; P(2, — 2, 2); u = Duf = i + - 2|3 k

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Q: 3. Evaluate f 2xdx + (y + 2)dy along the rectangle with vertices (0, 0), (0,3), (3,3) and (0, 3).

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Q: Calculate fz(3, 6, 9), where f(x, y, z) = xyz. fz(3, 6, 9) =

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Q: d) Let g(x,y) = (3x + y)2(2y + 1) and h(x, y) = 6y2(2x + y) h i) Find дх’ ду’ дх’ ду

A: The given functions are  and

Q: 3x + 11y af If f(x, y) = then is : 3x – 11y dy - 66x (3x + 11y)² 66x (3x – 11y)² 66x (3x + 11y)2 66x…

A: Partial derivative

Q: Evaluate S(z² + 3z)dz along the straight line from (2,0) to (2, 2) and then from (2, 2) to (0, 2).

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Q: Determine D,f (4,-1,6) for f(x,y,z) = ery + 4zx in the direction of (-3,2,6).

A: Given fx,y,z=ex2y+4zx3 in the direction of -3,2,6

Q: linearise R = AT + BT

A: To linearise the below equation. R=AT+BT2

Q: Find those constants a, b, c, d such that f(z) = ax² + bxy + y² + i(x² + cxy + dy?)

A: The given function is : fz=ax2+bxy+y2+i(x2+cxy+dy2)

Q: ▼ × (-y + x, x² + 3zy, zxy³) =? O (3y - 3zx², zx³, −2+x) O(-3y + 3xy²z, -y³z, 1+ 2x) ○ (2y + x²z,…

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Q: Find f,(2, 4) and f,(2, 4) for f(x, y) = x* + 4x?y³ + 8y*. 2.3 f-(2, 4) = f,(2, 4) =

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Q: ts) Find two linearly independent solutions of 2x²y" - xy + (2x + 1)y=0, x > 0 e form - 1/1

A: for the differential equation 2x2y''-xy'+(2x+1)y=0,x>0 To find the ser

Q: r(2,4) Sa (2y+x) dx +(3x– y)dy} along the straight line x-2t and Evaluate J(0,3) y=P+3.

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Q: f(x, y) = y² - 6xy + 10x² Which points lie on: f(x,y) = 5? (1,1), (1,5), (2,9)

A: In this question, we will check which points lie on f(x,y)= 5.