Ordinary differential equations questions.

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#	Question
1	State the order, degree and linearity of the ordinary differential equation given here under. Verify that the given solution satisfies the equation $x`=x^2 +1, x=tan(t+k), kin mathbb{R}$ . Mathematical Calculation
2	Given the ordinary differential equation $x`=0.5x$ , with the general solution $x=ke^{0.5t}$ and initial condition x(0)=0. Find the value of k and sketch the graph of x over t. Mathematical Calculation
3	Use the method of variation of parameters to find the solution of the equation $y"-5y`+6y=2e^t$ Mathematical Calculation
4	Solve the initial value problem $frac{d^{2}y}{dx^{2}}+lambda ^{2}x=A_0sinalpha t$ , $x(0)=x`(0)=0$ where $A_0$ is a constant and $alpha eq lambda$ . Mathematical Calculation
5	Determine and classify the singular points of the following differential equation $(x^{4}-2x^{3}+x^{2})y"+2(x-1)y`+x^{2}y=0$ Mathematical Calculation
6	Find a series solution of the equation , $y"-y=0$ about $x_0=0$ .. Mathematical Calculation
7	Use the Laplace transforms to find the solution of the differential equation $y"-3y`+2y=e^{-4t}$ , $y(0)=1, y`(0)=5$ . Mathematical Calculation
8	Find a fundamental set of solutions of $x`=egin{pmatrix} -5 & 1\ 4& -2 end{pmatrix}x, x(0)=inom{1}{2}$ and draw a phase potrait for the system. Mathematical Calculation
9	Solve the following IVP and find the interval of validity of the solution $frac{dy }{dx}=e^{-y}(2x-4), y(5)=0$ $frac{dy }{dx}=6y^{2}x, y(1)=frac{1}{25}$ $y`=frac{xy^{3}}{sqrt{1+x^{2}}}, y(0)=-1$ $frac{dr}{d heta}=frac{r^{2}}{ heta}, r(1)=2$ Mathematical Calculation
10	Find the solution and interval of validity to the following IVP $frac{2ty}{t^{2}+1}-2t-(2-ln(t^2+1))y`=0$ $y`+frac{4}{x}y=x^{3}y^{2}, y(2)=-1, x>0.$ Mathematical Calculation