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Differential Equations question

2016 · Shift 0 · Q25
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  5. /2016 · Shift 0 · Q25

Differential Equations question

2016 · Shift 0 · Q25

JEE MainMathematicsDifferential EquationsMCQ+4 / −1
If a curve y=f(x)y=f(x)y=f(x) passes through the point (1,−1)(1,-1)(1,−1) and satisfies the differential equation, y(1+xy)dx=xdyy(1+xy) dx=xdyy(1+xy)dx=xdy, then f(−12)f\left( { - {1 \over 2}} \right)f(−21​) is equal to :
  1. A
    25{2 \over 5}52​
  2. B
    45{4 \over 5}54​
  3. C
    −25-{2 \over 5}−52​
  4. D
    −45-{4 \over 5}−54​
View written solutionFree

Correct answer: B

  1. Given differential equation

We have y(1+xy) dx=x dy.y(1+xy)\,dx = x\,dy.y(1+xy)dx=xdy. So, xdydx=y(1+xy).x\frac{dy}{dx} = y(1+xy).xdxdy​=y(1+xy). Hence, dydx=yx+y2.\frac{dy}{dx} = \frac{y}{x} + y^2.dxdy​=xy​+y2.

  1. Use a substitution

Since the equation contains yx\dfrac{y}{x}xy​ and y2y^2y2, let v=1y.v=\frac{1}{y}.v=y1​. Then dvdx=−1y2dydx.\frac{dv}{dx} = -\frac{1}{y^2}\frac{dy}{dx}.dxdv​=−y21​dxdy​. Using dydx=yx+y2,\frac{dy}{dx} = \frac{y}{x}+y^2,dxdy​=xy​+y2, we get

= -\frac{1}{xy}-1.$$ Since $v=\frac{1}{y}$, $$\frac{1}{xy}=\frac{v}{x}.$$ Therefore, $$\frac{dv}{dx} + \frac{v}{x} = -1.$$ 3. **Solve the linear differential equation** The equation is $$\frac{dv}{dx} + \frac{1}{x}v = -1.$$ Its integrating factor is $$\text{I.F.}=e^{\int \frac{1}{x}dx}=x.$$ Multiplying throughout by $x$: $$x\frac{dv}{dx}+v=-x.$$ So, $$\frac{d}{dx}(xv)=-x.$$ Integrating, $$xv=-\frac{x^2}{2}+C.$$ Thus, $$v=-\frac{x}{2}+\frac{C}{x}.$$ Since $v=\frac{1}{y}$, $$\frac{1}{y}=-\frac{x}{2}+\frac{C}{x}.$$ 4. **Use the condition $(1,-1)$** At $x=1$, $y=-1$, so $$\frac{1}{y}=-1.$$ Substitute into the solution: $$-1=-\frac{1}{2}+C.$$ Therefore, $$C=-\frac{1}{2}.$$ So the solution becomes $$\frac{1}{y}=-\frac{x}{2}-\frac{1}{2x} = -\frac{x^2+1}{2x}.$$ Hence, $$y=-\frac{2x}{x^2+1}.$$ 5. **Find $f\left(-\frac12\right)$** Substitute $x=-\frac12$: $$y=-\frac{2\left(-\frac12\right)}{\left(-\frac12\right)^2+1} =\frac{1}{\frac14+1} =\frac{1}{\frac54} =\frac{4}{5}.$$ So, $$f\left(-\frac12\right)=\frac45.$$ 6. **Check options** The correct option is: - **B: $\frac{4}{5}$**
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