1-An expression of the differential form 𝑀(𝑥,𝑦)𝑑𝑥+𝑁(𝑥,𝑦)

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1-An expression of the differential form 𝑀(𝑥,𝑦)𝑑𝑥+𝑁(𝑥,𝑦)𝑑𝑦 is an
exact differential in a region R of the xy-plane if there is a
function u(x,y) such that 𝜕𝑢𝜕𝑥=𝑀 and 𝜕𝑢𝜕𝑦=𝑁. The total differential
of u satisfies 𝑑𝑢=𝜕𝑢𝜕𝑥𝑑𝑥+ 𝜕𝑢𝜕𝑦𝑑𝑦=𝑀𝑑𝑥+𝑁𝑑𝑦. If M(x,y)dx + N(x,y)dy is
an exact differential form, then the 𝑀(𝑥,𝑦)𝑑𝑥+𝑁(𝑥,𝑦)𝑑𝑦=0 equation
is called an exact equation.
Solve for the equation of 3𝑥2+2𝑦𝑑𝑦𝑑𝑥=0, for when 𝑦(0)=4. Include
text of exactness. Show each step of your calculation in
detail.
2-Solve the equations below by using the Second Order
Homogeneous equation. Show each step of your calculation in
detail.
a) 𝑦′′+3𝑦=0 when 𝑦(0)=1 and 𝑦′(0)=3
3-By using the Method of Undetermined Coefficient for the Second
Order Non Homogeneous equation solve for 𝑦′′+3𝑦=18𝑥2 when 𝑦(0)=−3
and 𝑦′(0)=0. Show each step of your calculation in detail.