Differential equations practice · Easy

Option C ($\frac{{d^2 y}}{{d x^2}}=0$) is correct because the general equation of a line is y = mx + c, and its second derivative with respect to x is always zero.

Option D (-e²logₑ(2)) is correct because it is the value of x(e) obtained by solving the given differential equation with the initial condition x(1) = 0.

Option D (None of These) is correct because after simplification, the given equation has only two independent arbitrary constants, making the order of the differential equation 2.

Option B (3) is correct because the general equation of a parabola whose axis of symmetry is parallel to the x-axis has three arbitrary constants, thus the differential equation is of order 3.

Option A, $(2,-2)$, is the correct ordered pair $(k, l)$ obtained by finding the differential equation of the given family of curves.

Option B ($m=3, n=2$) is correct because the order of a differential equation is the highest order derivative present, and the degree is the power of the highest order derivative after the equation has been made polynomial in derivatives.

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

Option D (a parabola with vertex origin and axes coincide with $x$-axis for $n=2$) is correct because the given condition leads to the equation $y^n = Cx$, which represents a parabola when $n=2$ with vertex at the origin and axis along the x-axis.

Option B ($y^{\prime} y^{\prime \prime}+y=\sin x$) is correct because it contains a second derivative ($y^{\prime \prime}$) and no higher-order derivatives.

Option A is correct because it represents the area of the intersection of two circles, calculated using geometry and trigonometry.

Option B ($\cot x$) is correct because if $\sin x$ is the integrating factor, then $P$ must be the derivative of $\log(\sin x)$, which is $\cot x$.

The correct answer is C ($y=-K x^3 / 2+C x$) because it satisfies the differential equation derived from the given condition about the y-intercept of the tangent.

Option A (2,2) is correct because, after squaring twice to remove the square roots, the highest order derivative is 2 and its power is 2.

Option D (-2) is correct because it is the value obtained by integrating the given differential equation and using the provided conditions to find the constants.

Option A (0) is correct because the two distinct solutions of the differential equation, given the initial conditions, never intersect, meaning they are parallel lines in a transformed space.

Option D (P=5, Q=-6, R=3) is correct because these values satisfy all the given conditions for the function f(x).

Option D (Exists and equals 4) is correct because by solving the given differential equation and finding the limit, we obtain a value of 4.

Option A, $\sin \left(\frac{y}{x}\right)=\log x+\frac{1}{2}$, is correct because it is the solution to the given differential equation that passes through the specified point.

Option C ($x y=C$) is the correct solution because it is obtained by integrating the given differential equation and simplifying the result.

Option B ($y=(1 / 2) x+C x^2$) is correct because it is the curve that satisfies the given condition about the trapezoid area.

Option C ($\frac{{4+e^2}}{{4}}$) is correct because it is the value of y(e) obtained by solving the given differential equation with the initial condition y(2)=2.

Option A (5) is correct because after simplifying the summation, we get $\tan^{-1} \frac{20}{21}$, so m+n = 41 and $\left[\log_2 41\right] = 5$.

The correct answer is B ($y=C(x^2+y^2)$) because it represents the family of curves orthogonal to the given trajectories, found by solving the differential equation.

Option B (family of ellipses) is correct because the solution to the given differential equation represents an ellipse.

Option C (1,2) is correct because the degree of the differential equation is 1 and the order is 2.

Option C (2,1) is correct because the order is the highest derivative (2) and the degree is the power of the highest derivative (1).

Option B (4) is correct because solving the differential equation and applying the given conditions leads to the equation of the curve, which allows us to find the value of |y(1/8)|.

Option B ($\frac{{1}}{{4}} e^{-2 x}+c x+d$) is the correct solution obtained by integrating the given second-order differential equation twice.

Option A is correct because it is the solution obtained by solving the given differential equation using the substitution method.

Option A ($y^k=C x$) is correct because it is the solution obtained by integrating the differential equation formed from the given condition about subtangents.