Find the general solution of the separable differential equation: dy/dx = 2xy.
Answer and explanation
Correct answer: B. y = C * e^(x^2)
- Step 1: Separate variables: (1 / y) dy = 2x dx.
- Step 2: Integrate both sides: integral (1 / y) dy = integral 2x dx.
- Step 3: Compute: ln|y| = x^2 + C_1.
- Step 4: Exponentiate: |y| = e^(x^2 + C_1) = e^(C_1) * e^(x^2).
- Step 5: Let C = +- e^(C_1): y = C * e^(x^2).
Answer
y = C * e^(x^2).
Why not the others: 'y = C e^(2x)' treats x as a constant during integration; 'y = x^2 + C' ignores 1/y separation; 'y = C ln(x^2)' inverts exponentiation.
Practise more Calculus and Analytical Geometry.
Free questions with an explanation for every one, and full timed Qatar Senior Secondary Certificate (Thanaweya) mock tests.
More Calculus and Analytical Geometry questions
- Evaluate the limit: lim_(x -> 0) [sin(5x) / (2x)].
- Find the derivative dy/dx of the implicit equation x^3 + y^3 = 6xy at the point (3, 3) on the Folium of Descartes.
- What is the equation of the normal line to the curve y = x^2 - 4x + 5 at the point (1, 2)?
- Evaluate the definite integral: integral from 0 to 1 of [x / (x^2 + 1)] dx.
- Find the absolute maximum value of the function f(x) = 2x^3 - 3x^2 - 12x + 1 on the closed interval [-2, 3].
- What is the total area of the finite region enclosed between the parabola y = 4 - x^2 and the line y = 0?