Suppose y = y(x) be the solution curve to the differential equation
\(\frac{dy}{dx}−y=2−e^{−x}\) such that \(\lim_{{x \to \infty}}y(x)\)
is finite. If a and bare respectively the x – and y – intercepts of the tangent to the curve at x = 0, then the value of a – 4b is equal to _____.
Suppose y = y(x) be the solution curve to the differential equation
\(\frac{dy}{dx}−y=2−e^{−x}\) such that \(\lim_{{x \to \infty}}y(x)\)
is finite. If a and bare respectively the x – and y – intercepts of the tangent to the curve at x = 0, then the value of a – 4b is equal to _____.
Correct Answer: 3
Solution and Explanation
The correct answer is 3
If \(= e^{−x}\)
\(y⋅e^{−x} =−2e^{−x}+\frac{e^{−2x}}{2}+C\)
\(⇒y=−2+e^{−x}+Ce^x\)
\(\lim_{{x \to \infty}}\) y(x)is finite so C=0
y = –2 + e–x
\(⇒\frac{dy}{dx}=−e^{−x}\)
\(⇒\frac{dy}{dx}| =−1\)
Equation of tangent
y + 1 = –1 (x – 0)
or y + x = –1
So a = –1, b = –1
⇒ a–4b = 3
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations