Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
MATCH THE FOLLOWING:
(A) If the curve satisfy the equation $(e^x+1)ydy = (y+1)e^x dx$ passes through $(0,0)$ and $(k,1)$ then $k$ is
(B) If the curve satisfy the equation $x\frac{dy}{dx}+y=xy^3$ passes through $(1,1)$ and $(\frac{3}{2},p)$ then $p$ is
(C) If $\frac{dy}{dx}=\frac{xy+y}{xy+x}$ then the solution of the differential equation always passes through the point origin and $(k,1)$ then $k$ is
(D) The solution of the equation $\frac{dy}{dx}=\frac{3x-4y-2}{3x-4y-3}$ passes through origin the distance of it from $(-1,1)$ is
Step-by-Step Solution
Key Concept: Solve four distinct differential equations using variable separation, Bernoulli substitution, homogeneous equation transformation, and substitution methods respectively, then use initial conditions to find specific parameter values or geometric properties.
Part (A) solves $y(e^x + 1)dy = (y+1)e^x dx$ by separating variables and integrating to obtain $(e^x + 1)(y + 1) = cxe^x$. Part (B) transforms $x\frac{dy}{dx} + y = xy^3$ using substitution $t = \frac{1}{y^2}$ to get a linear equation, solved as $\frac{1}{2x^2y^2} + \frac{1}{x} = c$. Part (C) solves $\frac{dy}{dx} = \frac{xy + y}{xy + x}$ by separating variables after recognizing the homogeneous structure, yielding $\log(\frac{y}{Ax}) = x - y$. Part (D) substitutes $3x - 4y = X$ to reduce a nonlinear equation and integrates to find $-X + 4\log(X+1) = x + c'$.
Correct Answer: [A-r] [B-s] [C-p] [D-q]