Please help me solve this simple ODE!!

Please give me a generous step-by-step solution to this attached ODE, thanks

Solve the ODE.

Solve the following ODE : x'' + x = cos(t)

Nullclines, Direction Field Analysis, Equilibrium Points and Phase Portrait

Consider the following second order system: x'=y y'=x+xy a. Find the nullclines and do direction field analysis on them. b. Find the equilibrium points of this system and classify them... c. Sketch a phase portrait. Please see the attached file for the fully formatted problems.

Differentiation of Standard Functions (4 Problems)

Differentiate each of the following functions with respect to the independent variable, use the above worked examples as an example: Please see the attached file for the fully formatted problems.

Determine if the following system has nay non-constant solutions that are bounded, i.e. do not run off to infinity in magnitude x’ = x(y – 1) y’ = y(x – 1) Explain in some detail, the reason for your answer.

Ordinary Differential Equation Determine if the following system has nay non-constant solutions that are bounded, i.e. do not run off to infinity in magnitude x’ = x(y – 1) y’ = y( ...continues

Systems of Differential Equations : Hamiltonian Systems

Consider the following system x'=-2y y'=x/2 Show that this system is a Hamiltonian system. Please see the attached file for the fully formatted problem.

Third Order Taylor Polynomial Approximation of a Solution to an ODE

Find a third order Taylor polynomial approximation, centered at t=0 of the solution to: x'' + tx = 0 x(0) = 0 x'(0) =0 Please see attached.

Ordinary Differential Equations : Boundary Value Problems

Please see the attached file for the fully formatted problems. Solve each of the following ODE's....

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Determine if the following equation is exact. If it is, solve it. If not, try to solve it by finding an integrating factor. cosx + y(sinx)y'=0

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Use variation of parameters to solve the differential equation y'''+4y'=cot2t.