Differential equations practice · Mixed difficulty

Option C (-5) is correct because solving the differential equation and applying the initial condition yields y(π) = -5.

Option A ($x\frac{{{d^2}y}}{{d{x^2}}} + 2\frac{{dy}}{{dx}} - xy = 0$) is correct because it represents the differential equation obtained after differentiating the given equation twice and eliminating the arbitrary constants a and b.

Option C (${\left( {\sqrt e } \right)^{{{\left( {\log x} \right)}^2}}}$) is correct because it is the integrating factor obtained by solving the integral of the coefficient of y after rewriting the differential equation in standard form.

Option A ($2,2$) is correct because after squaring both sides to remove the fractional exponent, the highest order derivative is $\frac{{d^2y}}{{dx^2}}$ (order 2) and its power is 2 (degree 2).

Option B ($x$) is correct because it is the integrating factor of the given differential equation after rearranging it into the standard form.

Option A ($\frac{{{d^2}y}}{{d{x^2}}} = 0$) is correct because the general equation of a non-vertical line is $ax + by = 1$, and its second derivative with respect to x is zero.

Option C ($\frac{e}{4}$) is correct because it's the value of y(e) obtained by solving the given differential equation and using the initial condition.

Option C ($2y{e^x} = {e^{2x}} + c$) is correct because it is the solution to the linear differential equation obtained by finding the integrating factor.

Option B ($\sin^{-1} x + \sin^{-1} y = C$) is correct because it is the result of separating variables and integrating the given differential equation.

Option B (2) is correct because the order of a differential equation is the highest order derivative present in the equation, which is the second derivative in this case.

Option A is correct because solving the differential equation and applying the initial condition yields the given function.

Option A (2) is correct because the general equation of such a parabola contains two arbitrary constants.

Option A ($\log \left(\frac{y}{x}\right)=c x$) is the correct solution to the differential equation, obtained by substituting $y=vx$ and integrating.

The correct answer is D ($y=2 \sqrt{{c x}}+c^2$) because it satisfies the given differential equation when $p = \sqrt{{\frac{{c}}{{x}}}}$ is substituted.

Option A (4) is correct because the area is calculated by integrating the function y = sqrt(36x) from x = 0 to x = 1.

Option C (A is true, R is true, $\mathrm{R} \Rightarrow \mathrm{A}$) is correct because both the integrating factor for the given equation and the general formula for the integrating factor are correct, and the general formula implies the specific case.

Option C ($(y-x \frac{{dy}}{{dx}})^2=1+(\frac{{dy}}{{dx}})^2$) is correct because it represents the differential equation for the family of lines at a unit distance from the origin, derived by eliminating the parameter from the general equation of such lines.

Option B ($y = \frac{{{x^2}}}{2}\log x - \frac{{{x^2}}}{4} + c$) is correct because it is the solution to the given differential equation obtained by integrating by parts.

Option A (${\tan ^{ - 1}}(\frac{x}{y}) + \log y + c = 0$) is correct because it is the general solution obtained by solving the homogeneous differential equation.

Option B ($y=\left(c_1+c_2 x\right) e^x$) is correct because it represents the general solution to the given second-order homogeneous linear differential equation with repeated roots.

Option B (-$\frac{{\pi}}{{6}}$) is correct because it's the value of y(-$\sqrt{{3}}$) obtained after simplification and substitution into the given equation.

Option B (2) is correct because the highest order derivative present in the differential equation is the second derivative.

Option B ($y\cos x = {e^x} + c$) is correct because it is the solution obtained by finding the integrating factor and solving the linear differential equation.

Option B ($1+\frac{{1}}{{y}}-\frac{{e^{1 / y}}}{{e}}$) is correct because it is the solution to the given differential equation with the specified initial condition $y(1)=1$.

Option A ($2 \ln 18$) is correct because it represents the time calculated by solving the given differential equation with the initial condition and setting the population to zero.

Option A ($x^2+1$) is correct because it is the integrating factor obtained by solving the differential equation.

Option B ($x^3 y+3 e^x=3 C y$) is correct because it is the general solution obtained by integrating the given differential equation after rewriting it in an integrable form.

Option D ($y = \tan \left( {x + \frac{{{x^2}}}{2}} \right)$) is correct because it is the solution to the given differential equation with the initial condition y(0) = 0.

Option B (${e^y} - {e^x} = \frac{1}{3}{x^3} + c$) is correct because it is the solution to the given separable differential equation.

Option C ($\frac{{1}}{{(x+y)^2}}=x^2+1+c e^{{x}^2}$) is correct because it represents the solution obtained by substituting $v = x+y$ and solving the resulting differential equation.