Which of the following statements are true? L tfen bir ya da daha fazlas?n? se'e7in: The equation y = 5e^{3t} is a solution of the differential equation 4y'' - 8y' + 3y = 0. The differential equation xy' - y sin x + y^5 = 0 is Bernoulli equation. dy/dx = xe^{x^2 - ln(y^2)} is not separable differential equation. Every solution of the differential equation dy/dt = y - 5 can be written in the form y = 5 + ce^t for some constant c. The equation y' = sin(x^2 - 1)y / (y^2 - 1) is separable. The differential equation (d^2y/dx^2)^3 + 2dy/dx = x^2 sin(x + y) is a second order nonlinear equation. e^{t^2}y'(t) + e^{sin(t)}y(t) = 0 is a nonlinear differential equation.
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The equation y = 5e^(3t) is a solution of the differential equation 4y'' + 8y' + 3y = 0. To check if this is true, we need to find the first and second derivatives of y and plug them into the given differential equation. y(t) = 5e^(3t) y'(t) = 15e^(3t) y''(t) = Show more…
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Which of the following statements are true? Do not try to solve the differential equations explicitly: (a) Consider the differential equation y' = 1/(1+x^4) y + cos(3x) Depending on the initial condition y(0), the solution may not exist for all values of x (blow up in finite x). (b) If y(x) is a solution of the differential equation y' = (1 - x)y^2, then for any real number c, cy(x) is also a solution of the same equation. (c) If y1(x) and y2(x) are solutions of the differential equation y'' + e^(-5x)y' + sin(2x)y = ln(1 + x^2), then y1(x) + y2(x) is also a solution of the same equation. (d) The functions 1/(1+x^4), (1 - x), and sin(2x) are linearly independent.
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Which of the following statement(s) are true for the given differential equation? x^2y'' - 5xy' + 13y = 30x^2 Lütfen bir ya da daha fazlasını seçin: r^2 - 5r + 13 = 0 is a characteristic polynomial of the differential equation. Its homogeneous solution is y_H = e^{3z}(C_1 cos(2x) + C_2 sin(2x)) for arbitrary constants C_1 and C_2. It is a second order nonhomogeneous Cauchy-Euler differential equation. To find particular solution, the variation of parameters can be used. Its characteristic polynomial has 3 + 2i and 3 - 2i complex roots. Using t = ln x, we get d^2y/dt^2 - 6dy/dt + 13y = 30e^{2t}.
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