Question

Ord Verify that y - ecos 2x is a solution to " -6y +13yo Verify thaty is a solution to y 2x o. Show your work! Solve the following Differential Equations: (SHOW YOUR WORK!!) xa+(cos x)y y(0) 10 Use integrating factor or separable method dy v exy2 Use Bernoulli's Method y(0) Use Substitution method with: ux -sin(x+ y da y )

Answer 1

"Let y""-6y' + 13y = 0-(1) be given differential equation let y = e^3x cos 2x y' = e^3x (-sin 2x)(2) + e^3x(3)cos 2x therefore y"" = -2e^3xcos 2x(2)-2e^3x(3)sin 2x + 3e^3x(-sin 2x)(2) + 3e^3x(3)cos 2x y"" = -4e^3x cos 2x-6e^3x sin 2x -6e^3x sin 2x + 9e^3x cos 2x therefore y"" = -4e^3x cos 2x - 6e63x sin 2x -6e^3x sin 2x + 9e^3x cos 2x consider lhs = y""-6y' + 13y = (5e^3xcos 2x-12e63xsin 2x)-6(-2e^3xsin 2x? + 3e^3x cos 2x) + 13(e^3x cos 2x) = 5e^3x cos 2x-12e^3xsin 2x + 12e^3x sin 2x-18e^3x cos 2x + 13e63x cos 2x = 0 = rhs therefore y = e^3x cos 2x is a solution of differential equation y""-6y' + 13y = 0 \r\n let y'-2xy^2 = 0-(1) let be given differential equation let y = 1/(4-x^2) therefore y' = (4-x^2)d/dx(1)-(1) d/dx(4-x^2)^2 therefore y' = (4-x^2)(0)-(4-2x)/(4-x^2)^2 therefore y' = 2x/(4-x^2)^2-(2) also y^2 = (1/4-x^2)^2 = 1/(4-x^2)^2 -(3) consider lhs = y'-2xy^2 = 2x/(4-x^2)-2x(1/(4-x^2)^2)"