3(a) Critically explain with practical examples on how a particular i solution of a differential equation can be deduced from the general solution. (b) Find the equation of the tangent line to the curve 8x²y3 – 5x² +7y³ = 10(x– 3y), at the point (2,-2). (c) Given T" = 6x²y³ – cosx+ 5y² – siny + 10x² – e-3*; determine (1) T' = [(T")ôx (ii) T' = S(T")ay (d) Solve the equation cosecxcosydy– 10xdx = 0, separably.

Q: Consider a nonhomogeneous differential equation y" + 4y = -12x + 10e-*. (a) Verify that yp = -3x +…

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Q: 5. The standard form of the first order linear differential equation dy-(cscx-ycotx)dx=0is

A: Given differential equation, dy-cscx-ycotxdx=0 The objective is to find the standard form of the…

Q: Find the differential equation of the cubics : cy^2= x^2 ( x - a ) with a held fixed.

A: Here c is arbitrary constant and a  is fixed. So we have to find a differential equation which does…

Q: 1. Find real-valued fundamental solutions of the differential equation 1 x' = Ax, A= 2 %3D 4 -3

A: We need to find the solutions of given differential equations. Given - X' = AX, where A = 1-24-3 So,…

Q: 2. Obtain the Differential Equation from the given general solution. y = Cj cos(3x) + C2 sin(3x) +…

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Q: An ordinary differential equation of a horizontal string which is freely pinned at both ends is…

A: Given, d2ydx2-a2y=w2Elx2-lx So, the characteristics equation of the given equation is, m2-a2=0m=±a…

Q: 3. Find solution of the differential equation; x "- 5x'+ 6x 6t-4 using Laplace transforms for the…

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Q: 1. Verify that for constant C the function ø(x) = + Cx is a solution of the differential equation 2…

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Q: 1. Given the differential equation: y" = sin(5x). Determine its general solution. A. 150y = cos(5x)…

A: Given differntial equation          y'''=sin 5xthen we have to find it's general solution…

Q: 4. Find a third-order linear homogeneous differential equation which has the function r(t) = 3e-t +…

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Q: The results of an initial valued problem of a differential equation is a family of curves. (A) True…

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Q: x²(a+x) 2. Family of strophoids with the equation y² %3D a-x

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Q: Find a general solution of the second-order nonhomogeneous differential equation y" - 2y' + y =…

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Q: A solution for the differential equation is y = eux A1 sin (vx) + A2 cos (vx), where m and n are…

A: We will find out the required values.

Q: 1. Draw and obtain the differential equation of a circles having its center on the line x =y and…

A: 1.

Q: Solve the differential equation [cos (2y) — 3x²y²] dx + [cos (2y) - 2x sin (2y) - 2x³y] dy - a. sin…

A: Given: cos2y-3x2y2dx+cos2y-2xsin2y-2x3ydy=0

Q: Do the functions y₁ = cos3t and y₂ = sin3t constitute a fundamental set of solutions of the…

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Q: 4. Solve the differential equation 2x(2x – 1)- (x + 1)+y = 0 using the method of Frobenius. dx2…

A: 2x2x-1d2ydx2 +x+1dydx +y=0                       eq.(1) To solve by Frobenius series.

Q: 5, a) Solve the differential equation dy x+2y-1 %3D dx 2x-y+3

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Q: 1) Find the differential equation using method of undetermined coefficient y-3y' +2y e*sinx Aotoi…

A: Given: - The differential equation is, y''-3y'+2y=exsinx To find: - The solution of the given…

Q: Example 9: Obtain the differential equation of the co-axial circles of the svstem x² +y² + 2ax+c² =…

A: Answer:

Q: 11. Obtain the differential equation of the family of parabolas y =x +c and sketch those members of…

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Q: Consider the differential equation y' = 1/y - y Find its critical values, and classify them as…

A: Given,y'=1y-yCritical values:1y-y=0y2=1y=±1fy=1y-y∂f∂yy=±1=-1-1=-2<0So, both y=-1 , -1 are…

Q: State where in the ty-plane the hypothesis of the (non-linear) Existence and Uniqeness Theorem is…

A: Given: The given differential equation is y'=t-y2t+5y

Q: Form the differential equation for the function x+2y=a sin3x-bcos3x

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Q: Do the functions y = cos5t and y, = sin5t constitute a fundamental set of solutions of the…

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Q: The Solution of the differential equation g + 9y= -6 cos (3x) – 30 sin(3x) initial conditions…

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Q: Consider the differential equation 5y with a particular solution y = f(x) dæ having an initial…

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Q: An ordinary differential equation of a horizontal string which is freely pinned at both ends is…

A: Given problem is :

Q: Solve the Cauchy-Euler differential equation on the interval (0, ∞). x³y"" + 6x²y" + 4xy' - 4y = 0

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Q: Find a linear homogeneous differential equation with constant coefficients so that the general…

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Q: Consider the differential equation: (5x – 1)'y" (x) + 3(5x – 1)y(x) + y(x) = 9(5x - 1)-1 (1) a)…

A: The given differential equation is the Cauchy Euler differential equation.  To find the solution we…

Q: How do you solve the third-order differential equation y'''-3y''+2y'=3cos^2 2x using the Method of…

A: Consider the given equation. y'''-3y''+2y'=3cos22x Find the charectestic equation. m3-3r2+2r=0

Q: Solve the non-homogeneous differential equation of 6y"-y'-y=1-x-x?

A: Given differential equation is  6y"-y'-y=1-x-x2

Q: '+y- y²(cos 2x– sin 2x) = 0

A:

Q: y"+4y=x² sin 2 u

A: See the attachment

Q: 2. Solve the differential equation using undetermined coefficients satisfying the given conditions.…

A: Given differential equation  y"+y'=sin x    ;  y(0) =y'(0) =0 As per our guideline, we are supposed…

Q: Determine whether Theorem 1.2.1 guarantees that the differential y= √y2-9 equation possesses a…

A: To Determine -

Q: Determine the form of a particular solution of the differential equation y" + 9y = e*cos3x + x² +…

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Q: Consider the nonhomogeneous second order differential equation a°y" – xy' + y = nT +0, x > 1.

A: First we have to find the general solution of the differential equation x2y''-xy'+y=xlnx where x≠0,…

Q: Solve the differential equation 3y + 3y (sin-¹x)√1-x² : 1

A: The given differential equation is 3y'+3ysin-1x1-x2=1.

Q: Determine the solution of the first-order ordinary differential equation: 2y' +4y- 4y cos x 0

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Q: Determine whether the Theorem 1.2.2 guarantees that the differential equation y' = Vy2 – 9 possesses…

A: We have to solve given problem:

Q: 5) Find a homogeneous linear differential equation with constant coefficients whose general solution…

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Q: In a spring mass system the displacement, x meters, of an object from its equilibrium position with…

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Q: Consider the differential equation a²y" – 3xy' + 3y 2.a* sin(x) whose complementary equation has…

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Q: For the differential equation y' = 2y - x, draw line segments representing the slope field at the…

A: Given that, y' = 2y - x dy/dx = 2y - x At the point (1, 2) we get: dy/dx = 2(2) - 1 dy/dx = 4 - 1…

Q: Consider a nonhomogeneous differential equation y" + 4y = -12x + 10e. (a) Verify that yp = -3x + 2e*…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.