1. Identify the order and degree of the differential equations: 1.a. Given equation: $ (y'')^{-2} + y' = xy'' + \sin x $ - Order: The highest derivative is $y''$, which is the second derivative, so order = 2. - Degree: The equation contains $(y'')^{-2}$, which is a negative power, so the degree is not defined (degree must be a positive integer). 1.b. Given equation: $ y'' = (\sqrt{1 + y'})^{\frac{3}{2}} $ - Order: Highest derivative is $y''$, order = 2. - Degree: The right side involves fractional powers of $y'$, so degree is not defined. 1.c. Given equation: $ (y''')^2 + (y'')^3 = \cos x $ - Order: Highest derivative is $y'''$, order = 3. - Degree: The powers of derivatives are integers (2 and 3), so degree = 3 (highest power of highest order derivative). 2. (a) Find the Wronskian of $y_1(t) = e^{kt}$ and $y_2(t) = te^{kt}$: - Wronskian formula: $$W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'$$ - Compute derivatives: $y_1' = ke^{kt}$ $y_2' = e^{kt} + kte^{kt} = e^{kt}(1 + kt)$ - Substitute: $$W = e^{kt} \cdot e^{kt}(1 + kt) - te^{kt} \cdot ke^{kt} = e^{2kt}(1 + kt) - kte^{2kt} = e^{2kt}$$ 2. (b) Given $W(y_1,y_2) = x^2 e^x$ and $y_1(x) = x$, find $y_2(x)$: - Wronskian formula: $$W = y_1 y_2' - y_2 y_1'$$ - Substitute known values: $$x^2 e^x = x y_2' - y_2 \cdot 1 = x y_2' - y_2$$ - Rearrange: $$x y_2' - y_2 = x^2 e^x$$ - Divide by $x$ (assuming $x \neq 0$): $$y_2' - \frac{1}{x} y_2 = x e^x$$ - This is a linear ODE for $y_2$ with integrating factor: $$\mu = e^{-\int \frac{1}{x} dx} = e^{-\ln x} = \frac{1}{x}$$ - Multiply both sides by $\frac{1}{x}$: $$\frac{1}{x} y_2' - \frac{1}{x^2} y_2 = e^x$$ - Left side is derivative of $\frac{y_2}{x}$: $$\frac{d}{dx} \left( \frac{y_2}{x} \right) = e^x$$ - Integrate both sides: $$\frac{y_2}{x} = \int e^x dx = e^x + C$$ - Solve for $y_2$: $$y_2 = x(e^x + C) = x e^x + C x$$ Final answers: - 1.a: Order = 2, Degree = not defined - 1.b: Order = 2, Degree = not defined - 1.c: Order = 3, Degree = 3 - 2.a: $W = e^{2kt}$ - 2.b: $y_2 = x e^x + C x$