This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — I will solve the three Bernoulli equations shown on the whiteboard. The general form of a Bernoulli equation is (dy)/(dx) + P(x)y = Q(x)y^n. 1) Solve x (dy)/(dx) + y = x^2 y^2 Step 1: Rewrite the equation in standard Bernoulli form. Divide by x: (dy)/(dx) + (1)/(x)y = xy^2 Here, P(x) = (1)/(x), Q(x) = x, and n=2. Step 2: Apply the substitution u = y^1-n. For n=2, u = y^1-2 = y^-1 = (1)/(y). This implies y = u^-1. Differentiate y with respect to x: (dy)/(dx) = -u^-2 (du)/(dx). Step 3: Substitute y and (dy)/(dx) into the equation. -u^-2 (du)/(dx) + (1)/(x)u^-1 = x(u^-1)^2 -(1)/(u^2) (du)/(dx) + (1)/(xu) = (x)/(u^2) Step 4: Convert to a linear first-order differential equation. Multiply by -u^2: (du)/(dx) - (u)/(x) = -x This is a linear equation of the form (du)/(dx) + P_1(x)u = Q_1(x), with P_1(x) = -(1)/(x) and Q_1(x) = -x. Step 5: Find the integrating factor (IF). IF = e^ P_1(x) dx = e^ -(1)/(x) dx = e^-|x| = e^|x^-1| = x^-1 = (1)/(x) Step 6: Multiply the linear equation by the integrating factor. (1)/(x) (du)/(dx) - (1)/(x^2)u = -x · (1)/(x) (d)/(dx)((1)/(x)u) = -1 Step 7: Integrate both sides with respect to x. (d)/(dx)((1)/(x)u) dx = -1 dx (1)/(x)u = -x + C Step 8: Solve for u and substitute back u = (1)/(y). u = -x^2 + Cx (1)/(y) = Cx - x^2 Step 9: Solve for y. y = (1)/(Cx - x^2) The final answer is y = (1)/(Cx - x^2). 2) Solve (dy)/(dx) + xy = xy^3 Step 1: Identify the components of the Bernoulli equation. The equation is already in the standard form (dy)/(dx) + P(x)y = Q(x)y^n. Here, P(x) = x, Q(x) = x, and n=3. Step 2: Apply the substitution u = y^1-n. For n=3, u = y^1-3 = y^-2 = (1)/(y^2). This implies y = u^-1/2. Differentiate y with respect to x: (dy)/(dx) = -(1)/(2)u^-3/2 (du)/(dx). Step 3: Substitute y and (dy)/(dx) into the equation. -(1)/(2)u^-3/2 (du)/(dx) + x u^-1/2 = x(u^-1/2)^3 -(1)/(2)u^-3/2 (du)/(dx) + x u^-1/2 = x u^-3/2 Step 4: Convert to a linear first-order differential equation. Multiply by -2u^3/2: (du)/(dx) - 2x u^(3/2 - 1/2) = -2x (du)/(dx) - 2xu = -2x This is a linear equation with P_1(x) = -2x and Q_1(x) = -2x. Step 5: Find the integrating factor (IF). IF = e^ P_1(x) dx = e^ -2x dx = e^-x^2 Step 6: Multiply the linear equation by the integrating factor. e^-x^2 (du)/(dx) - 2x e^-x^2 u = -2x e^-x^2 (d)/(dx)(e^-x^2u) = -2x e^-x^2 Step 7: Integrate both sides with respect to x. (d)/(dx)(e^-x^2