1. a) Solve the following differential equations: ydx x i) -1 (tan-¹x) dy = ydx x² +1 (In x)dy, b) Find the orthogonal trajectories of the family of curves y = x+k ii) xy' = y + √√√x² - y².

Q: (2xy + 3y²) dx – (2xy + x²) dy = 0 .. [VARIABLE SEPARABLE/HOMOGENEOUS |

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Q: 4. (6 points) Using separation of variables, solve the differ- ential equation, (1+7)dy dx y Use C…

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Q: 2. Solve for y in terms of x for the following equations: a) In(1- 2y) = x %3D b) In(y- 1) - In 2 =…

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Q: the derivatives of y with respect to x: y = xe* – ex y = e(1x+) %3D y = e*(sin x + cos x) %3D y =…

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Q: Consider the finction f(x, y) = 6y² + 4x² – 12xy- 4y. Then: (1,1) is a critical point and f has a…

A: The given function is f(x,y)=6y2+4x2-12xy-4y

Q: b. Using differential equations, find the orthogonal trajectories of the following family of curves:…

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Q: b. (1+y²)y" + xy' - 3y = cos x solve and show that this equation is non- linear and non-homogeneous.…

A: As per the question we have to use the solution of the given second order differential equation to…

Q: v = ) V In(2x-3y

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Q: 6. Solve these equations y' (t) - ay(t) = Qet, starting from y(0) = 2: y'(t) − y(t) = 8e-3t y'(t) +…

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Q: Solve Q11

A: 11. Given that the series                      x∑m=1∞mamxm-1+∑m=0∞amxm

Q: Q.No.3: a) Solve Cauchy-Euler equation d'y 3x3 dx3 d'y + 9x2 + 6x dx2 dy 6y = 1+ x3 dx -

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Q: a) 12 Q.7) To write the power series £=3 4m as the form -4m a) m - 1 we must shift m to become : d)…

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Q: Solve the following first order equations: (e" + y)dx + (2+x + ye")dy = 0, y(0) = 1.

A: First order linear differential equation

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A: Given- The equation 80x'' + 1000x = 2500cosπvt6 To find- FInd the general solution to this…

Q: 1. Find a general solution of the first order equation: y'= 3x²y².

A: The given first order differential equation is solved by method of separation of variable.

Q: c) y(x) = c, + cz + Acos3x + Bsin3x Q.6) The singular point (s) of (x² + 4)y" – x y' + 4y = 0 , is…

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Q: Find a particular equation for the DE: y+4y'-2y= 2x -3x+6

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Q: 7. Find an equation of the line tangent to the curve y = 3x² – 1 and perpendicular to the line x

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Q: 11. Solve: 2xyy'= x² y° +4y² a. 8x' = y(C – x*) b. 8r = y (C – x*) c. x² = y(C – 8x*) d. 8x = y*(C…

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Q: Find the differential dy of the function y = 4x²-x+ 3 Select one: O a. (8x-1)dx O b. (x²-x² + 3x) dx…

A: Solve for the differentiation