Differential equations — practice questions

Step 1: Identify $P(x)$ and $Q(x)$. The equation is in the form $\frac{dy}{dx} + P(x)y = Q(x)$. Here, $P(x) = -\frac{2}{x}$ and $Q(x) = x^2$.

Step 2: Calculate the Integrating Factor (IF). $I(x) = e^{\int P(x) dx} = e^{\int -\frac{2}{x} dx} = e^{-2\ln|x|} = e^{\ln|x^{-2}|} = x^{-2} = \frac{1}{x^2}$.

Step 3: Multiply the DE by the IF. $\frac{1}{x^2}\frac{dy}{dx} - \frac{2}{x^3}y = \frac{x^2}{x^2}$ The left side becomes $\frac{d}{dx}(\frac{1}{x^2}y)$. So, $\frac{d}{dx}(\frac{y}{x^2}) = 1$.

Step 4: Integrate both sides with respect to $x$. $\int \frac{d}{dx}(\frac{y}{x^2}) dx = \int 1 dx$ $\frac{y}{x^2} = x + C$, where C is the constant of integration.

Step 5: Rearrange to find the general solution. $y = x^2(x+C) = x^3 + Cx^2$.

Step 1: Find the Complementary Function (CF). The auxiliary equation is $m^2 + 5m + 6 = 0$. Factoring gives $(m+2)(m+3)=0$, so the roots are $m_1 = -2$ and $m_2 = -3$. The CF is $y_{CF} = Ae^{-2x} + Be^{-3x}$.

Step 2: Find the Particular Integral (PI). The right-hand side is $12e^{-x}$. This form is not in the CF. Try a PI of the form $y_{PI} = Ce^{-x}$. Then $\frac{dy_{PI}}{dx} = -Ce^{-x}$ and $\frac{d^2y_{PI}}{dx^2} = Ce^{-x}$. Substitute into the original DE: $(Ce^{-x}) + 5(-Ce^{-x}) + 6(Ce^{-x}) = 12e^{-x}$ $Ce^{-x} - 5Ce^{-x} + 6Ce^{-x} = 12e^{-x}$ $2Ce^{-x} = 12e^{-x}$ $2C = 12 \implies C = 6$. So, the PI is $y_{PI} = 6e^{-x}$.

Step 3: Form the General Solution. $y = y_{CF} + y_{PI} = Ae^{-2x} + Be^{-3x} + 6e^{-x}$.

Step 4: Use initial conditions to find A and B. We have $y(0)=1$ and $y'(0)=0$. First, find $\frac{dy}{dx} = -2Ae^{-2x} - 3Be^{-3x} - 6e^{-x}$. Using $y(0)=1$: $1 = Ae^0 + Be^0 + 6e^0 \implies 1 = A + B + 6 \implies A+B = -5$ (Eq 1). Using $y'(0)=0$: $0 = -2Ae^0 - 3Be^0 - 6e^0 \implies 0 = -2A - 3B - 6 \implies 2A+3B = -6$ (Eq 2).

Solving simultaneously: Multiply (Eq 1) by 2: $2A+2B = -10$. Subtract this from (Eq 2): $(2A+3B) - (2A+2B) = -6 - (-10) \implies B = 4$. Substitute $B=4$ into $A+B=-5$ to get $A+4=-5 \implies A=-9$.

Step 5: Write the particular solution. $y = -9e^{-2x} + 4e^{-3x} + 6e^{-x}$.