2. Find the solution of the given initial value problem. y’’+2y’+2y=0, y(π/4)=2, y’(π/4)=-2 topic: complex roots of the characteristic equation.
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2. Find the solution of the given initial value problem.
y’’+2y’+2y=0, y(π/4)=2, y’(π/4)=-2
topic: complex roots of the characteristic equation.
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- Find two linearly independent solutions of y″+2xy=0 of the form and enter the first few coefficients shown in the photoWe have shown that in the case of complex conjugate roots,r =a±ib,b̸=0, of the indicial equation, the general solution to the Cauchy-Euler equation y′′+a1y′+a2y =0, x > 0 is y(x)=xa[c1 cos(b ln x)+c2 sin(b ln x)]. .........................(8.8.26) Q. Show that (8.8.26) can be written in the form y(x)= Axa cos(b ln x −φ) for appropriate constants A and φ.C.1. Give a real general solution to the equationu'''' - 5u''' -6u'' = 0 : C.2. Give a real general solution to the equation(D2 + 8D + 41)^3 (D + 7)^2w = 0 : C.3. Give a real general solution to the equationy'' - 36y = 8e^2t ;given that (-1/4e2t) is a solution to it. C.4. Give a real general solution to the equationv'''' + 13v'' + 36v = 10e^t ;given that (1/5e^t) is a solution to it.
- B. NONHOMOGENEOUS EQUATIONS: UNDETERMINED COEFFICIENTS 2. (D² + D – 2)y = 2x – 40cos2x 3. (D² + 1)y = sinxII. Consider x4 + 4 = 0. a. Find all solutions of this equation in Z4. b. Find all solutions of this equation in Z[i].Determine the appropriate form for a particular solution of the fifth-order equation(D - 2)3 (D2 + 9)y = x2e2x + x sin 3x.
- 4.3.5 Exercises Finding Particular Solutions. Find a particular solution for each equation in Exercise Group 4.3.5.1-10 using complexification. 1. " + 4y = 3 cos 2t 2. dz TZ dr2 dt 56+7u 7. y Foy+by 3. 21 9. - 2 cos 2t 3 sin 2t cos 3t y" + 7y² + 10µ--4sin 3t Hint Acoume the complex solution h has formu. Ae3it 4. " 25 SCOST 6 8. 10. "A 13y 9cos 26 +2 +32 2 sin 2t da2 dt will Avl 20ucos 5t u" +4u' + 20u = -3 sin 3t Find a par- Finding Frequencies, Amplitudes, and Phase Angles. ticular solution of the form Up = A cos(wto) for each equation in ExerciseFind a general solution to the following higher-order equations. (a) y'"-5y'' + 5y + 11y=0) (b) y + 5y + 4y' - 10y = 0 (c) y + 20y" + 64y = 0 *** (a) What are the roots of the auxiliary equation? The roots are -1,3+ i√√2,3-i√√2. (Type an exact answer, using radicals and i as needed. Use a comma to separate answers as needed.) What is the general solution? y(t) = (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)3 Consider the Cauchy-Euler equation x" + t2 0 = x (a) What is the associated indicial equation? Use m as the variable. Your answer should be an equation. (b) Find the general solution to the Cauchy-Euler equation. Use ci and c2 as arbitrary constants. x(t) %3D
- Describe your solution to the problem: find the general solution to y′′+3y=ex using the easiest method. List all the other possible methods to solve the problem.3. Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution. Do not solve the equation. Explain what method might work. (a) y" – 2y + 5y = 3 tan t + et (b) y" – 2y + 5y = (e2i + 1)²A4 Put the equation xy" + 2y′ + (x³ + λx²) y = 0 into the Sturm-Liouville form (py')' + qy + λoy = 0, stating what p, q and σ are.