Advanced engineering mathematics by Peter V O'Neil

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By Peter V O'Neil

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5. y − 2y = −8x 2 In each of Problems 6 through 10, solve the initial value problem. 6. y + 3y = 5e2x − 6; y(0) = 2 7. y + 1 x−2 y = 3x; y(3) = 4 8. y − y = 2e4x ; y(0) = −3 9. y + 10. y + 2 x+1 5y 9x y = 3; y(0) = 5 = 3x 3 + x; y(−1) = 4 11. Find all functions with the property that the y intercept of the tangent to the graph at (x, y) is 2x 2 . 12. A 500 gallon tank initially contains 50 gallons of brine solution in which 28 pounds of salt have been 5 gal/min; 1/2 lb/gal Tank 1 13. 6. Tank 1 initially contains 20 pounds of salt dissolved in 100 gallons of brine.

There is a simple test to determine whether M + N y = 0 is exact for (x, y) in a rectangle R of the plane. 1 Test for Exactness Suppose M, N , ∂ N /∂ x and ∂ M/∂ y are continuous for all (x, y) in some rectangle R in the (x, y)− plane. Then M + N y = 0 is exact on R if and only if ∂N ∂M = ∂x ∂y for (x,y) in R. Proof If M + N y = 0 is exact, then there is a potential function ϕ and ∂ϕ ∂ϕ = M(x, y) and = N (x, y). ∂x ∂y Then, for (x, y) in R, ∂M ∂ = ∂y ∂y ∂ϕ ∂x = ∂ 2ϕ ∂ 2ϕ ∂ = = ∂ y∂ x ∂ x∂ y ∂ x ∂ϕ ∂y = ∂N .

2 x Upon multiplying by −2v 3/2 , we obtain the linear equation 2 v − v = −6x 2 . x This has integrating factor e −(2/x) d x = eln(x −2 ) = x −2 . Multiply the differential equation by x −2 : x −2 v − 2x −3 v = −6. This is (x −2 v) = −6, and an integration yields x −2 v = −6x + c. Then v = −6x 3 + cx 2 . In terms of y, the original Bernoulli equation has the general solution 1 1 y(x) = √ =√ . 3 The Riccati Equation A differential equation of the form y = P(x)y 2 + Q(x)y + R(x) is called a Riccati equation.

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