<p><strong>Suppose y = y(x) be the solution curve to the differential equation <math><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>−</mo><mi>y</mi><mo>=</mo><mn>2</mn><mo>−</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow></math> such that <math><mrow><munder><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></munder><mi>y</mi><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math> is finite. If a and b are respectively the x and y intercepts of the tangent to the curve at x = 0, then the value of a 4b is equal to…………</strong></p>

dydx−y=2−e−xI.F.=e−∫dx=e−x∴ solnye−x=∫ (2e−x−e−2x)dxy=−2+e−x2+Cexas for x ∞ y finite c = 0∴y=e−x2−2⇒x+2y=−3⇒a=−3 b=−32∴a=4b=−3+6=3

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Suppose y = y(x) be the solution curve to the differential equation $\frac{d y}{d x} - y = 2 - e^{- x}$ such that $\underset{x \to \infty}{l i m} y (x)$ is finite. If a and b are respectively the x and y intercepts of the tangent to the curve at x = 0, then the value of a 4b is equal to…………

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alok kumar singh | Contributor-Level 10

2 months ago

$\frac{d y}{d x} - y = 2 - e^{- x}$

$I . F . = e^{- \int d x} = e^{- x}$

$∴$ soln

$y e^{- x} = \int (2 e^{- x} - e^{- 2 x}) d x$

$y = - 2 + \frac{e^{- x}}{2} + C e^{x}$

as for x $\infty$ y finite c = 0

$∴ y = \frac{e^{- x}}{2} - 2$

$\begin{array}{l} \Rightarrow x + 2 y = - 3 \\ \Rightarrow a = - 3 b = \frac{- 3}{2} \end{array}$

$∴ a = 4 b = - 3 + 6 = 3$

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