Solutions of Differential Equations Complete #1-10. Prove that the following are solutions to the given differential...

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Solutions of Differential Equations Complete #1-10. Prove that the following are solutions to the given differential equations, using the example below as a guide. You must show all of your work, using proper notation. Example: You will want to balance the differential equation by finding y' using the solution equation and plugging that in for the y' on the left side and then plug in the value of what y equals using the given the solution on the right to prove that both sides balance. Differential equation: y' = 5y y' = Ce5x.5 y' = 5Ce5x y' = 5y 5Ce5x = 5y 5Ce5x = 5Ce5x Solution: y = Ce 5x Differential equation Solution 1. 3y' + 5y = e2x = - y = e2x Differential equation y' = 2xy x - y2 Solution x+ y = Cy 3. Differential equation dy dx = y2 - 1 Solution y - 2 ln(y) = x Differential equation 4. y" + y = 0 Solution y = C sin(x) - C2 cos(x) Differential equation 5. y" + 2y +2y=0 Solution y= Ce*cos(x) + 2e* sin(x) Differential equation Solution 6. y" + y = tan(x) y= = cos(x)In sec(x) + tan(x) Differential equation 7. y" + 4y = 2ex y = 25 Solution (e4x + e*) Solutions of Differential Equations Complete #1-10. Prove that the following are solutions to the given differential equations, using the example below as a guide. You must show all of your work, using proper notation. Example: You will want to balance the differential equation by finding y' using the solution equation and plugging that in for the y' on the left side and then plug in the value of what y equals using the given the solution on the right to prove that both sides balance. Differential equation: y' = 5y y' = Ce5x.5 y' = 5Ce5x y' = 5y 5Ce5x = 5y 5Ce5x = 5Ce5x Solution: y = Ce 5x Differential equation Solution 1. 3y' + 5y = e2x = - y = e2x Differential equation y' = 2xy x - y2 Solution x+ y = Cy 3. Differential equation dy dx = y2 - 1 Solution y - 2 ln(y) = x Differential equation 4. y" + y = 0 Solution y = C sin(x) - C2 cos(x) Differential equation 5. y" + 2y +2y=0 Solution y= Ce*cos(x) + 2e* sin(x) Differential equation Solution 6. y" + y = tan(x) y= = cos(x)In sec(x) + tan(x) Differential equation 7. y" + 4y = 2ex y = 25 Solution (e4x + e*)