Ordinary differential equations questions.

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#	Question
1	Solve the cauchy euler differential equation $x^{2}y'-2xy`+2y=0$ Mathematical Calculation
2	Solve the initial value problem $frac{dy}{dx}=frac{y^{2}}{x},,,y(1)=2$ And find the interval of validity of the solution. Mathematical Calculation
3	Solve the differential equation $6y`-2y=xy^{4},,,y(0)=-2.$ Mathematical Calculation
4	If $y_{1}$ and $y_{2}$ are linearly independent solutions of $t^{2}y'- 2y`+(3+t)y=0$ And if $W(y_{1},y_{2})(1)=2$ , find the value of $W(y_{1},y_{2})(4)$ . Mathematical Calculation
5	Consider the initial value problem $y'+2y`+6y=0,,,,, y(0)=2, y`(0)=alpha geq 0$ Find the solution $y(t)$ of this problem Find $alpha$ so that y=0 when t=1. Find as a function of $alpha$ , the smallest positive value of t for which y=0. Determine the limit of the expression found in part (iii) as $alpha ightarrow infty$ Mathematical Calculation
6	Determine whether the given pair of functions is linearly independent or linearly dependent. $f(t)=t^{2}+5t,,g(t)=t^{2}-5t$ $f( heta)=cos,3 heta,,g( heta)=4,cos^{3}, heta-3,cos, heta$ $f(t)=e^{{lambda} t}cos,mu t,,g(t)=e^{{lambda} t}sin,mu t,,mu eq 0$ Mathematical Calculation
7	Solve $x"+3x`+2x=30e^{2t}$ Use undetermined coefficients method Use variation of parameters method Mathematical Calculation
8	Reduce the Cauchy-Euler differential equation $2x^{2}y"+3xy`-y=0$ to constant coefficients second order homogenous differential equation. Mathematical Calculation
9	Check the exactness and solve the given differential equation $(3x^{2}-y^{2})dy-2xydx=0$ Mathematical Calculation
10	Use the Laplace transform to find the solution of the following system $left{egin{matrix} y`+x=1\ x`-y=-4e^t & x(0)=0,&y(0) =0 end{matrix} ight.$ Mathematical Calculation