4) The vertical displacement of a spring mass system from its natural length is described by dy Laplace transform, solve the differential equation of the mass after t = 20 seconds if the initial position is y(0) = 0 m/s and the initial velocity is y'(0) = 0 m/s. dy + 2+ 2y = e-t where time t is measured in seconds. Using dt2 dt

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

A:

Q: Use the slope field for the differential equation y′ = 1/x, where x > 0, to sketch the graph of…

A: Given the differential equation y’ = 1/x, where x > 0, we have to sketch the graph of the…

Q: Suppose that x1(t) = et and x2(t) = e-t are two solutions of a homogeneous linear differential…

A: The answer is found in step-2

Q: The differential equation for a mass spring system without damping is modeled by y" + 2y = 0. Find…

A: Let's find power series.

Q: Heat transfer from a body to its surroundings by radiation, based or he Stefan-Boltzmann law, is…

A:

Q: Write down a differential equation of the form dt all other solutions diverge from y = 4 as t →∞.…

A: Let us consider a differential equation dydt=ay+b  such that the solution diverges as t→∞, y=C The…

Q: from the partial differential equation by eliminating the arbitrary function z=(x2+y2+z2)

A:

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

A:

Q: What is of the differential equation (3 +y+ 2y sin² æ) dx + (x + 2xy – ysin2x) dy = 0 ? O 2x sin 2x…

A:

Q: A forced, damped, mass-spring system satisfies the differential equation a" + 4x' + 5x = 3 cos(t) +…

A: Given that x''+4x'+5x=3cost+sint Take x''+4x'+5x=0 Assume m=ddt, m2+4m+5=0m=-4±16-202m=-2±i So the…

Q: 9. The characteristics of the partial differential equation *3 = 0 ду a əx dy +2- дх ду + cos? when…

A:

Q: At any given temporal coordinate, the rate of change at which the vertical position of an object…

A: Given that, Vertical position of an object is governed by the differential equation given below…

Q: By using Laplace transform to solve y" + y = e with y (0) y (0) = y" (0) = 0, what would be Laplace…

A:

Q: 3. Plot the slope field for the differential equation = y² – x and sketch the solution for the…

A: Given: A differential equation dydx=y2-x. To sketch: Slope field of given differential equation and…

Q: The motion of the mass-spring system is governed by the differential equation d²y dt2 dy + 6 dt + 9y…

A: The motion of the mass-spring system is given by the differential equation: d2ydt2+6dydt+9y=cos(2t)…

Q: A nomal is drawn at a point P(x, y) of a curve, it meets the x-axis at Q. If PQ is of constant…

A: Equation of normal at the point P(x,y) to the curve is Y-y=-dxdy(X-x). ⇒X=x+ydydx Thus,…

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

A:

Q: By applying Newton's second law to a body at a certain distance x at time t, the following…

A:

Q: Find the general solution of the differential equation = 6 tan(5y) using separation of variables. dx…

A:

Q: after using z = x-y as substitution, the DE (x-y)^2 * dy/dx = a^2 (a is constant) transforms to what…

A:

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: Eliminate the arbitrary functions and hence obtain the partial differential equations:   z = f(x *…

A:

Q: y = - is one of the solutions to the non-homogeneous Cauchy-Euler equation 1 x²y" + xy' – y x + 1…

A:

Q: Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt…

A:

Q: A slope field shows that the slope at the point (1, 1) is 6. Does this slope field represent the…

A: Given yr = 4x +2y

Q: An electric circuit, consisting of a capacitor, resistor, and an electromotive force can be modeled…

A: Please see below pictures for detailed solution.

Q: The given differential equation is already in standard form, as the coefficient of the highest order…

A:

Q: Sketch the direction field of the differential equation y'=cos(xy); trace a solution through the…

A:

Q: A population (in thousands) grows according to the differential equation dy e ky(L-y), where the…

A:

Q: The differential equation governing the velocity of an object is dv = -kv", dt where k > 0 and n are…

A: The given differential equation is dvdt=-kvn, where k>0 and n are constants and v(0)=v∘. At t=0,…

Q: The origin is the only critical point of the nonlinear second-order differential equation x" + (x)²…

A:

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: Consider the nonlinear time-dependent ordinary differential equation (ODE) y '(t) = y ^2 (t)+…

A:

Q: Suppose you have a first order differential equation of the form: M(x, y) + N(x, y)y' = 0 Further…

A:

Q: Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt…

A: To Determine -

Q: The bellow differential equation represents a projectile motion of a body where g is a constant of…

A:

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

A:

Q: 2(a) Form the partial differential equation from z = ar + by + ab by eliminating the constants a and…

A:

Q: The formation of the partial differential equation by eliminating arbitrary constants a, b from the…

A:

Q: Q.2 (a) Form Partial differential equation from z = ax² + by?

A: the given equation is z = ax2 + by2 .......(1)

Q: Suppose y₁ = sinh 2x, y2 = 6x, y3 = cos 2x are solutions of a linear homogeneous differential…

A: To find the differential equation :-

Q: Suppose a spring with spring constant 6 N/m is horizontal and has one end attached to a wall and the…

A: Given that, the spring constant k=6 N/m, the mass m=3 kg and friction b=3 N.s/m. a. Obtain the…

Q: Suppose g: IR →R has 3 distinct roots. How many distinct piecewise-constant solutions does the…

A:

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

A:

Q: A heart pacemaker consists of a switch, a battery of constant voltage Eo, a capacitor with constant…

A:

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: A dynamic system is represented by the following differential equation. y" - y' - 2 sin y = u(t),…

A:

Q: If the differential equation +8. + 9x = m dt2 is overdamped, the range of values form is? Your…

A: Let the differential equation is,                       ad2xdt2 +bdxdt +cx =0 .............(1) It…

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.