Ordinary differential equations questions.

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#	Question
1	Solve the initial value problem and sketch the plane for the system. $left{egin{matrix} x_{1}^{`}=x_{1}+2x_{2}\ x_{2}^{`}=3x_{1}+2x_{2} end{matrix} ight.$ Given that x₁(0)=0 and x₂(0)=-4. Mathematical Calculation
2	Briefly explain the meaning of implicit and explicit solutions of a differential equation . Use the given differential equation to show the difference between implicit and explicit solution. $y`=frac{x}{y^{`}} ,, y(1)=2)$ Mathematical Calculation
3	Solve the following Bernoulli differential equation $2y`+(tan,x)y=frac{(4x+4)^{2}y^{3}}{cos,x}$ Mathematical Calculation
4	Use the method of variation to parameters to solve the IVP $(x^2-1)y``+4xy`+2y=frac{2}{x+1}, , y(0)=-1, , y`(0)=-5$ then state the interval of the validity $y_{1}=frac{1}{x-1},,y_2=frac{1}{x+1}$ Mathematical Calculation
5	i. Define the terms order, degree and linearity of differential equations State the existence and uniqueness theorem for a first order initial value problem $y`=f(x,y),,,y(x_0)=y_0.$ Suppose that $n eq 0$ or $n eq 1$ . Show that the transformation $v=y^{1-n}$ reduces the Bernoulli equation $frac{dy}{dx}+P(x)y=Q(x)y^{n}$ to a linear equation in V. Solve the following initial valua problem and find the interval of validity for the solution. $y`+frac{4}{x}y=x^{3}y^{2},,,y(2)=-1, ,, x>0$ Mathematical Calculation