Question:

Solution of $\frac {d^2y} {dx^2}=x\,e^x$ is y

Updated On: Jul 7, 2022
  • $e^x(x - 2) + C_1 x + C_2$
  • $e^x(x - 1) + C_1 x^2 + C_2x$
  • $e^x(x + 1) + C_1 x^2 + C_2x$
  • $e^x(x - 2) - C_1 x + C_2$
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The Correct Option is A

Solution and Explanation

We have, $\frac{d^{2}y}{dx^{2}} = xe^{x}$ Integrating $\left(i\right)$ by parts, we get $\frac{dy}{dx} = xe^{x} - \int e^{x}\,dx+C_{1} = xe^{x} - e^{x} + C_{1}$ Again integrating by parts, we get $y = xe^{x} - e^{x} - e^{x} + C_{1}x + C_{2}$ $\Rightarrow y = xe^{x} - 2e^{x} + C_{1}x+C_{2}$ $\Rightarrow y = e^{x} \left(x - 2\right) + C_{1}x + C_{2}$.
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Concepts Used:

Order and Degree of Differential Equation

The equation that helps us to identify the type and complexity of the differential equation is the order and degree of a differential equation.

The Order of a Differential Equation:

The highest order of the derivative that appears in the differential equation is the order of a differential equation.

The Degree of a Differential Equation:

The highest power of the highest order derivative that appears in a differential equation is the degree of a differential equation. Its degree is always a positive integer.

For examples:

  • 7(d4y/dx4)3 + 5(d2y/dx2)4+ 9(dy/dx)8 + 11 = 0 (Degree - 3)
  • (dy/dx)2 + (dy/dx) - Cos3x = 0 (Degree - 2)
  • (d2y/dx2) + x(dy/dx)3 = 0 (Degree - 1)