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: Strategic substitutions and variable separation transform complex differential equations into linear or separable forms.
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]