Homework Problems for Handout Sheet 02 1. Solve the BVP: (6xy-4)dx+(3x^2+2y)dy=0, y(2)=4. 2. Solve the DE: (4xy+3y^2-1)dx+(2x^2+6xy-4y)dy=0. 3. Solve the DE: (xe^x+6y)dx+(6x-e^y)dy=0. 4. Solve the DE: ((3-y)/x^2)dx+((y^2-2x)/xy^2)dy=0. 5. Solve the DE: (2x sin y-cos^2 y)dx +(2x sin y cos y+x^2 cos y+4y)dy=0. 6. Solve the DE: (y sec^2 x+sec x tan x)dx+(tan x+2y)dy=0. Answers: 1. 3x^2y-4x+y^2=56. 2. 2x^2y+3xy^2-x-2y^2=K. 3. xe^x-e^x+6xy-e^y=K. 4. y^2-3y+2x=Kxy. 5. x^2 sin y-x cos^2 y+2y^2=K. 6. y tan x+sec x+y^2=K.
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For problem 1, we have the BVP: $(6xy^4)dx + (3x^2 + 2y)dy = 0$ with the condition $9(2) = 4$. Show more…
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Homework Problems for Handout Sheet 02 1. Solve the BVP: (6xy-4)dx + (3x^2 + 2y)dy = 0, y(2) = 4. 2. Solve the DE: (4xy + 3y^2 - 1)dx + (2x^2 + 6xy - 4y)dy = 0. 3. Solve the DE: (xe^x + 6y)dx + (6x - e^y)dy = 0. 4. Solve the DE: ((3-y)/x^2)dx + ((y^2-2x)/xy^2)dy = 0. 5. Solve the DE: (2x sin y - cos^2 y)dx + (2x sin y cos y + x^2 cos y + 4y)dy = 0. 6. Solve the DE: (y sec^2 x + sec x tan x)dx + (tan x + 2y)dy = 0. Answers: 1. 3x^2y - 4x + y^2 = 56. 2. 2x^2y + 3xy^2 - x - 2y^2 = K. 3. xe^x - e^x + 6xy - e^y = K. 4. y^2 - 3y + 2x = Kxy. 5. x^2 sin y - x cos^2 y + 2y^2 = K. 6. y tan x + sec x + y^2 = K.
Sri K.
REVIEW PROBLEMS In Problems 1-30, solve the equation: 1. dy/dx = e^(x+y)/(y-1) 2. dy/dx - 4y = 32 3. (x^2 - 2y^-3)dy + (2xy - 3x^2)dx = 0 4. dy/dx + 3y/x = x^2 - 4x + 3 5. [sin(xy) + xy cos(xy)]dx + [1 + x^2 cos(xy)]dy 6. 2xy^3 dx - (1 - x^2)dy = 0 7. t^3y^2 dt + t^4y^-6 dy = 0 8. dy/dx + 2y/x = 2x^2y^2 9. (x^2 + y^2)dx + 3xy dy = 0 10. [1 + (1 + x^2 + 2xy + y^2)^-1] dx + [y^-1/2 + (1 + x^2 + 2xy + y^2)^-1] dy 11. dx/dt = 1 + cos^2(t - x) 12. (y^3 + 4e^xy)dx + (2e^x + 3y^2)dy = 0
Worksheet #1: First order linear differential equations Solve the following differential equations. You may need to bust out your integration note card! a) y' - 3y = e^x b) y' - 2xy = x c) y' + y = sin(e^x) d) y' = x + 5y e) y' + 4y = x f) xy' + 2y = e^{x^2}, x > 0 g) x^2y' + 2xy = cos x, x > 0 Solve the following initial value problems. a) y' + (cos x)y = cos x, y(0) = 2 b) y' + y = x + e^x, y(0) = 0 c) xy' = y + x^3 sin x, x > 0, y(pi) = 0 d) xy' - 3y = x^2, x > 0, y(1) = 0 e) y' - 2xy = 2xe^{x^2}, y(0) = 3 f) (1 + x^2)y' + 2xy = 3sqrt{x}, y(0) = 2
Adi S.
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