Answer Happy

I NEED ANSWERS TO ALL QUESTIONS.
Consider the initial value problem
y′+ (3/7)y=1-(1/3)t, y(0)=y0
Consider the initial value problem y′+73​y=1−31​t,y(0)=y0​. What equation expresses the requirement that the solution touches the t-axis? y′′(t)=0y(t)=0y′(t)=0​
What additional equation expresses the requirement that the solution does not cross the taxis? y′(t)=0y(t)=0y′′(t)=0​
The equations found in (a) and (b) form a system of two equations for t and y0​. Show how the DE can be used to solve directly for t. y′+73​y=0, but also 1−71​t=0, so t=7. y′+37​y=0, but also 1−31​t=0, so t=3. y′+73​y=0, but also 1−31​t=0, so t=3. y′+37​y=0, but also 1−71​t=0, so t=7.
Find the solution for the initial value problem with initial condition y(0)=y0​(0). y=(27112​)−97​t+y0​e−73​ty=(27112​)−97​t+(y0​−27112​)e−73​ty=(11227​)−499​t+(y0​+11227​)e−73​ty=(27112​)+349​t+(y0​−27112​)e−73​ty=−97​t+(y0​−27112​)e37​t​
Finish solving the equations found in (a) and (b). y0​=−499​e73​+27112​y0​=2749​e79​−27112​y0​=−2749​e79​+27112​y0​=27112​e499​+2749​y0​=−27112​e349​+2749​​