3. Given that f(x) = x is a solution of (1 – x² )y" – 2xy + 2y = 0 within -1

Q: 1. Solve the homogenous linear equation given below. xydx − (x + 2y)²dy = 0 F + ln (x+y) :) + 4 lny…

A: In this question we have to solve the homogenous linear equation.

Q: 1. Determine the general solution of the following nonhomogeneous linear equations. dy 1. dx ex 2+ y…

A: Second order linear non−homogeneous differential equation with constant coefficients A second order…

Q: Show that the equation x2 + y2 = 2003 has no solutions in theintegers.

A:

Q: 5. For the system of differential e rx + 2y 3x+ry figure out all the possible types of equilibrium…

A: The matrix form of the given system of differential equations is x^'=Ax^, with A=r33r, x^=xy.

Q: 5- The equation (y`-y =e2x ) is : ( linear, exact, separable, Bernoulli )

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Q: 7 Find the second linearly solution of the D.E x'y"+xy'-(x+2)y = 0, given that: y,(x) =

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Q: 4. Find the solution of the following set of simultaneous equations by LT with initial conditions…

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Q: Solve the IVP u® +u"=e*-2tx with y(0)=1, y'(0)=0, y"(0)=1 (where t is a real constant)

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Q: 6. Consider the equation y = 2y – 4t with initial condition y(0) using Euler's method with... = 0.…

A: Given: y'=ft,y=2y-4t a t0=0y0=0ft0,y0=0 By Euler method: yn+1=yn+hftn,ynwhere…

Q: Suppose we are given y₁) and y2(x) (with y₁ ‡ y2), which are two different solutions of a…

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Q: 16 Q4- Solve the following system of ODE dx dt Eaux"> dy dt = y + 1 =x+1

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Q: Consider the linear system : dx 2х — 2у dt dy - 2x + Y dt with initial conditions x(0) 0 and y(0) =…

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Q: Find the particular solution that satisfies the initial condition(s). y = C, + C2 In x xy" + y' = 0…

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Q: show that the solution y(t)=e^2tsin(t) is linearly independant.

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Q: 4. Find the solution of the following set of simultaneous equations by LT with initial conditions…

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Q: 1. Show that y = ze* + e-2* is a solution of y' + 2y = 2e*. 3

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Q: 2xyy'=1+y; whent = 2, y = 3 Ans. y = 5x-1

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Q: Find the number of equilibrium solutions of the r x' = x(y-2) โม่ = x − y

A: x'=x(y-2)y'=(x-y) Equilibrium solution of a system of differential equation X'=AX are those solution…

Q: 24. Find the particular solution that satisfies the initial condition. y'+y = 6e*, Initial…

A: The given equation is- y'+y=6ex Auxiliary equation of given equation is- D+1=0 Then, particular…

Q: 3. Find for the simplified general solution of (1 + x³)dy – x²ydx = 0 a. Iny =In(1 + x³) + C b. Iny…

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Q: The solution of y" + y = 0 with initial conditions y (0) = 1 and %3D y = 2 is

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Q: 1. (a) Show that (x) x² is an explicit solution to dy X- = 2y dx on the interval (-∞, ∞). (b) Show…

A: An explicit solution is any solution that is given in the form y=y(x)

Q: Solve the separable initial value problem. 1. y' = In(x)(1+y²), y(1) = 1 → y= 2. y =…

A: Use the following formulas 1.∫xndx=xn+1n+1+c2.∫11+x2dx=tan-1x+c3,∫ln x dx=xln x-x+c

Q: If yp=e4x is a particular solution of nonhomogeneous DE y’’-4y=12e4x and yp=x is a particular…

A: Note that if y1 is a particular solution of a nonhomogeneous ode y'' +p(x) y' +q(x)y=r(x) and y2 is…

Q: 6. Find the general solution to the system of equations dx dy + dt dt 2t + +x+y=0 d²x dx dt² dt

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Q: 2. What is the solution to the initial-value problem PD.E. ut = c²uxx (u(x,0) = 0 lu,(x,0) = xe-x -…

A: Given that  utt=c2uxx    -∞<x<∞,  0<t<∞ux,0=0  and  utx,0=xe-x2,   -∞≤x<∞ It is a…

Q: Find the general solution of the Cauchy-Euler equation: x²y" – 3xy + y = x°, x>0.

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Q: 5- Given that y₁ (t) = t is the solution to: + 17/12 y = ( 0 Use the reduction of order method to…

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Q: 2. Consider y+2-c y+1+cy, where e is a positive constant. For what values of e will the solution to…

A: Given yt+2-cyt+1+cyt

Q: 2 Discuss the stability of equilibrium points of the following ODE: { (2 = -r³ + y°, y = -x.

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Q: Put the following initial value problem into a first order system form. (Show your workl) y(0) – 3y"…

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Q: Verify that u(r, t) = f(e-"x) solves the initial value problem (au + z8,u = 0, u(x,0) = f(z).

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Q: 2. Use substitution to solve the second order y(0)=1 and linear IVP y"-y=0 with y'(0)=0. Hint: Let…

A: Given that, Second order differential equation is, y'' - y = 0…

Q: 8. Find the particular solution that satisfies the initial condition. y'+y = 6e*, Initial Condition:…

A: Find the particular solution

Q: 2. Find the first three approximations yı, y2, y3 using Euler's method for the initial value problem…

A: Let's find.

Q: Consider the initial value problem (x + 2) y" + with initial condition x 1 5Y' + e²y=x³ y (0) = 7 y'…

A: Given the Initial Value Problem x+2y"+1x-5y'+exy=x3 with given initial conditions. y0=7,y'0=2

Q: 1. The solution of (y - z) p + (z - x) q = x- y according to Lagrange's linear equation method is

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: 4. Find the solution of the following set of simultaneous equations by LT with initial conditions…

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Q: 9. Ify,(x)= sin 2r and y, (x) = cos 2x are two solutions of y"+ 4y 0, show that y, (x) and y, r) are…

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Q: Verify that the function y(r) = 2, is a solution of the initial value problem dy + 2y = 3y, v(0) =;…

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Q: yp=u1x+u2x^-2 is the particular solution proposed for the nonhomogeneous Cauchy-Euler equation…

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Q: 4. Use the method of undetermined coefficients to find a particular solution to the system below.…

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Q: 6. Find the particular solution of 2xydx + (y² – x²)dy = 0 ;where y = 2 and x =

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Q: 3. Find the general solution of the system: dx = x + y dt dy -2x – Y dt

A: Given dxdt=x+y and dydt=-2x-y

Q: Determine a general solution to the Cauchy-Euler equation z'y"(x) – 2y(z) = 3z² – 1, z>0.

A:

Q: Determine the solution of the simultaneous nonlinear equations and compute the errors. sin(xy) – x²…

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

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.