This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution P(t) = C $e^{0.2t}$. We then plug in the initial condition P(0) = 50 to find C = 50. This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it omits the constant determined by the initial condition, resulting in P(0) = 1 instead of 50. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution $y^2 = x^2 + C$. We then plug in the initial condition $y(0) = 3$ to find C = 9, giving $y = \sqrt{x^2 + 9}$. This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it omits the constant, resulting in $y(0) = 0 \neq 3$. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution y = C $e^{e^x}$. We then plug in the initial condition y(0) = 3 to find C = 3 $e^{-1}$, giving y = 3 $e^{e^x - 1}$. This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it ignores the constant adjustment, resulting in y(0) = 3 e ≠ 3. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution $y^2$ = $2x^2$ + C. We then plug in the initial condition y(1) = 2 to find C = 2, giving y = $√(2x^2$ + 2). This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it uses a negative constant, leading to imaginary values at x=1. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution B(t) = C $e^{3t}$. We then plug in the initial condition B(1) = 2 to find C = 2 $e^{-3}$, which gives B(t) = 2 $e^{3(t-1)}$. This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice A, fails because it uses the initial condition at t=0 instead of t=1, resulting in B(1) = 2 $e^{3}$ ≠ 2. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution y = C $e^{2x - x^2/2}$. We then plug in the initial condition y(1) = 3 to find C = 3 $e^{-3/2}$, giving y = 3 $e^{2x - x^2/2 - 3/2}$. This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it ignores the adjustment for the initial condition at x=1. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using separation of variables to solve the differential equation and then applying the initial condition to find the particular solution. By separating variables, we obtain dy/y = 3 dx/x, and integrating both sides gives ln|y| = 3 ln|x| + C, leading to the general solution y = K $x^3$. Substituting the initial condition y(2) = 12 into the general solution yields 12 = K cdot 8, so K = 3/2. This determines the constant K conceptually by solving the resulting equation after plugging in the given values. A tempting distractor is choice B, y = 12 $x^3$, which fails because it incorrectly uses the initial y-value as the coefficient without dividing by $x^3$ at the initial point. Always integrate the separated differential equation to find the general solution first, then substitute the initial condition to solve for the arbitrary constant.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution y = C $e^{x^2/2}$. We then plug in the initial condition y(0) = 5 to find C = 5. This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it omits the constant from the initial condition, resulting in y(0) = 1 instead of 5. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution y = C / $x^2$. We then plug in the initial condition y(2) = 1 to find C = 4, giving y = 4 / $x^2$. This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it halves the constant incorrectly, resulting in y(2) = 2/4 = 0.5 ≠ 1. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.

This problem requires using initial conditions to find the particular solution to a differential equation solved by separation of variables. After separating variables and integrating, we obtain the general solution $y^2$ = $x^2$ + 2x + C. We then plug in the initial condition y(0) = 1 to find C = 1, giving y = $√(x^2$ + 2x + 1). This determines the specific function that satisfies both the differential equation and the initial value. A common distractor, such as choice B, fails because it omits the constant, resulting in y(0) = 0 ≠ 1. In general, to apply initial conditions, substitute the given point into the general solution and solve for the constant to get the particular solution.