Differential equations practice · Mixed difficulty

Option A ($-\frac{\sin 3 x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3} x-1$) is correct because it is the solution to the given differential equation satisfying the initial conditions y'(0)=1 and y(0)=0.

Option B ($y \sec ^2 x=\sec x+c$) is correct because it is the solution obtained after finding the integrating factor and solving the linear differential equation.

Option B ($4 e^{3 x}-e^{-4 y}=3$) is incorrect. Option D ($3 e^{-4 y}+4 e^{3 x}=7$) is the correct solution, but it is not listed in the options. The explanation provides the correct solution, but it does not match any of the given options. The closest option is B, but it is not correct.

Option A ($y = {x^3} + 3x + 1$) is correct because it satisfies the given differential equation and the initial conditions.

Option D ($\frac{1}{x+1}$) is correct because it is the integrating factor obtained after converting the given differential equation to standard form and applying the formula for the integrating factor.

Option D ($2 y=e^{\operatorname{{Tan}}^{-1} x}+c e^{-\operatorname{{Tan}}^{-1}} x$) is incorrect. The correct answer is C ($2 y=e^{\operatorname{{Tan}}^{-1} x}+c e^{-\operatorname{{Tan}}^{-1}} x$). The error likely stems from incorrect handling of the integrating factor or the constant of integration.

Option A (1) is correct because the family of circles is defined by one parameter, leading to a first-order differential equation.

Option A ($y\frac{{dy}}{{dx}} = \pm \sqrt {{k^2} - {y^2}}$) is correct because it represents the differential equation relating the length of the normal to the y-coordinate.

Option A ($y''' - 6y'' + 11y' - 6y = 0$) is correct because it is the differential equation that satisfies the given function $y = A{e^x} + B{e^{2x}} + C{e^{3x}}$.

Option B ($\frac{3}{2}$) is correct because it is the value of y at x=2 after solving the given differential equation with the specified initial condition.

Option B ($x{(y')^2} = x + 2yy'$) is correct because it represents the differential equation obtained after eliminating the arbitrary constant from the given family of curves.

Option A ($y=\tan \left(\frac{{x+y}}{{2}}\right)+C$) is correct because it is the solution obtained by solving the given differential equation using a suitable substitution.

Option A ($y+\sqrt{{x^2+y^2}}=c x^2$) is correct because it is the general solution obtained by solving the given homogeneous differential equation.

Option A ($y\sec x = \tan x + c$) is correct because it is the solution to the given first-order linear differential equation after finding the integrating factor and integrating.

Option D is correct because after raising both sides to the power of 4 to remove the fractional exponent, the order is 2 (highest derivative is d²y/dx²) and the degree is 4 (power of the highest derivative).

Option B is correct because the differential equation is derived by differentiating the given equation twice and eliminating the arbitrary constants a and b.

Option B ($y=\tan \left(\frac{x^2}{2}+\cos x-1\right)$) is correct because it is the solution to the given differential equation with the specified initial condition.

Option A (3) is correct because the general equation of a parabola with directrix parallel to the x-axis has 3 independent constants, thus the order of the differential equation is 3.

Option D (30.1 years) is correct because it is the time it takes for the population to double at a 1% annual growth rate, calculated using exponential growth.

Option C ((-1/2, 4, 0)) is correct because it satisfies the given conditions: b = 2a and a is perpendicular to c.

Option B ($\sin x + \cos y = c$) is correct because it is the solution obtained by separating variables and integrating the given differential equation.

Option B ($y+\sqrt{x^2+y^2}=c x^2$) is the correct solution to the homogeneous differential equation, solved by substituting $y=vx$.

Option B ($\log \tan \frac{y}{2}+2 \sin x=\mathrm{C}$) is correct because it is the solution obtained by using trigonometric identities and integrating the differential equation.

Option B ($-\frac{1}{x y}+\log y=c$) is the correct solution obtained by separating variables and integrating the given differential equation.

Option A ($x=X+1, y=Y+0$) is correct because it is the substitution that reduces the given differential equation to a homogeneous form.

Option B ($y \frac{{d y}}{{d x}}+4 x=0$) is non-linear because it involves the product of the dependent variable $y$ and its derivative $\frac{{dy}}{{dx}}$.

Option C ($\sqrt {{{(1 - {x^2})}}}$) is correct because it is the integrating factor derived from the given differential equation after rewriting it in standard form.

Option B ($y=\frac{1}{18}(3 x^3+2 e^{3 x}-2)$) is correct because it satisfies the given differential equation and the initial conditions $y_1(0)=\frac{1}{3}$ and $y(0)=0$.

Option A, -3, is the correct value of k for which both statements are correct, as it satisfies both differential equations.

Option C (2,1) is incorrect. Option D (1,2) is incorrect. Option B (1,1) is correct because the order and degree of the differential equation are both 1 after simplification.