Express the solution of the given initial value problem in terms of a convolution integral y" + 12y' + 32y = cos(at); y(0) = 1, y'(0) = 0 y(t) = X+/ (--) - e-) cos(a 7) dr/ -2 e-8t +e-4t -4(t-T) -8(t-r)

Q: 5. Show that the differential equation y" +a²y = f(x), y(0) = y'(0) = 0, has 1 y(x) = -…

A: proof:-we have          y(x)=1a∫0xf(t)sina(x-t)dt   ⋯(i) differentiate by leibnitz's rule(i) , we…

Q: 3. Form the partial differential equation by eliminating the arbitrary constants a and b from z = ď²…

A:

Q: Show that ¢(x) = Ce³* + 1 is a solution to dy/dx – 3y = -3 for any choice of the constant C. Thus,…

A: We have to check Q(x)=Ce3x+1 is solution of given differential equation.

Q: 4) The vertical displacement of a spring mass system from its natural length is described by d²y dy…

A:

Q: What is the general solution of the differential equation y" +4y= tan 2x? Oy C1 cos 2x + C2 sin 2a -…

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Q: determine the general solution of the given differential equation that is valid in any interval not…

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Q: Solve the differential equatons x(2y + In ax) dy + (x+y)dx when x=3 y=Ź y= 2 1.

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Q: ди sin(x + t), at t = 0, 1, and when x = 0, ax u = 2t

A:

Q: 3. Form the partial differential equation by eliminating the arbitrary constants a and b from z =…

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Q: Show that y, = x* – x² Inx over the interval (0, 0) is the particular solution to the…

A: (C) Given differential equation is x2y''−4xy'+6y=2x4+x2....1 We have to find the particular solution…

Q: From the partial differential equation by elimanating the arbiraty function z=f(x2+y2+z2)

A: We have to form the partial differential equation.

Q: 1 of these 2 differential equations: dx dy O and (2x sin y +y) dx + (x² cos y + x)dy = 0 What are…

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Q: xydx -(x+2y)* dy

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Q: Determine a region of the ty-plane for which the given differential equation would have a unique…

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Q: 10x – 8yva? +1 fip dx 2 = 0.

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Q: Show that the solution of the differential equation (1–x*) = 2x(1+ y). dx 1+x given that y=1 when…

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Q: Show that y1 = sin x² and y2 = cos x² are linearly in- dependent functions, but that their Wronskian…

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Q: Find the value of x(0.4) for the coupled first order differential equations together with initial…

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Q: (a) Verify that 8x² - y? = c is a one-parameter family of solutions of the differential equation…

A: The differential equation ydydx=8x. We have to verify that 8x2-y2=c is a one-parameter family of…

Q: Find the solution of the differential equa­tion that satisfies the given boundary condition(s) . x"…

A: Provided differential equation is a homogeneous linear differential equation with constant…

Q: obtain in alinear differential eqaution with real, constant coefficients that is satisfied by the…

A: The solution of the problem is given in the next step.

Q: Eliminate arbitrary constant from z=(x-a^2)+(y-b^2) to from the partial differential equation

A: Given z=x-a2*y-b2Hence number arbitary constant is 2 and number of independent variable is 2 . So…

Q: (y') ² + (x-2x^-½) y'+ x^-1 - x^½= 0

A: As per the company rule, we are supposed to solve one problem. Therefore kindly repost the second…

Q: (b) Find the complete integral of non-linear first partial differential equation Oz. + y = qz.

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Q: Solve the initial value DE y' - ytanx = sinx, y(0) = 1 5-cos 2x 5-cos 3x 5-c A) y B) y = с) у %3D 4…

A: Given differential equation is y'-ytanx=sinx,   y0=1 Compare the given differential equation with…

Q: 3. Approximate the solution to the partial differential equation a²u ²u 5л 0x2 (x, y) + dyz (x, y) —…

A: Given: ∂2u∂x2x,y+∂2u∂y2x,y-12.5π2ux,y=-25π2sin5π2xsin5π2y To find: Dirichlet boundary condition

Q: This problem deals with the differential equation dx/dt kx(x – M) – h that models the harvesting of…

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Q: Consider the differential equation y = x+ y. Use Euler's method with Ax = 0.1 to estimate y when x =…

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Q: Suppose that x = 0 is a regular singular point of the differential equation x²y" – x(x + 3)y' + (x +…

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Q: 2(a) Given that f(x, y) = 6x²y3 – cosx + 5y – siny + 10x, establish the fact that, the order in…

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Q: Suppose that one solution y, (x) of the homogeneous second-order linear differential equation y"…

A: This can be evaluated as follows:

Q: The transversality condition holds for the partial differential equation X4x +Uy=0 with the initial…

A: In the given question we have to check the transversality  condition of the given P.D.E s.t xux+uy=0…

Q: Solve for the solution of the given Partial Differential Equations with boundary conditions. Əri =…

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Q: Find the integral surface passing through the curve 1 T: z = sin y + e' +- x = 0 8 of the partial…

A: In this question, we solve the partial differential equation by using the surface condition.

Q: 7: From an integral equation corresponding to the differential equation d²y dx² dy - sin x- +exy = x…

A: What is Integral Equation: Like differential equation, there is also integral equation. In integral…

Q: Suppose sinh 2x, y2 = 6x, y3 = cos 2x are solutions of a linear homogeneous differential equation…

A: To find the differential equation :-

Q: xy' + 2y' + 2y O with initial condition y(0) = 1

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Q: 3] Form the partial differential equation by eliminating arbitrary function from z = e® f(x – y) -

A: Given: Z=exfx-y

Q: 4) The vertical displacement of a spring mass system from its natural length is described by dy…

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

Q: Use centered finite differences to solve the ordinary differential equation for 0≤x≤6 d² u du +3…

A: We will use the basic knowledge of ordinary differential calculus to answer this question.

Q: Without solving explicitly, classify the critical points of the autonomous differential equation dx…

A: See the solution and follow the steps. You will understand easily.

Q: The solution of the differential equation by variable separable method dy (1+ y²) (3x² + 6) 5- is dx…

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Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

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Q: By using separation of variables u(x, y) = X(x)Y(y), break the partial differ- ential equation U =…

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Q: Form the partial differential equation by eliminating the arbitrary constants a and b from z = ď² x…

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Q: Find the particular solution of the differential equation (1-cos(2t))dy/dt=xsin(2t) given that…

A: The given differential equation is 1-cos2t dxdt=x sin2t     with xπ2=2. Solve as follows. 1-cos2t…

Q: Verify that the indicated function is an explicit solution of the given differential equation.…

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Q: Consider the differential equation dy dy +2- + 4y = 0; y(0) = 3, dx dy| = 0 dr dr² Is this initial…

A: First order forward finite difference is y'xi≈yxi+1-yxih   Second order forward finite difference is…

Q: Apply Runge-Kutta of 4th Order to approximate the solution on the interval [1,2] of the given…

A: Given: dydx=x-2sin2x-2xy, y1=2 To find: Approximate solution in the interval 1,2 with h=0.1.