Re-word_this_differential_equations_cheat_sheet_section_to_be_about@20241202_184412

user: Re-word this differential equations cheat sheet section to be about the complementary function then the particular function, instead of constant/non-constant coeffs (although still suggest what methods to use):

4. Second-Order Solutions:
   1. If constant coefficients, then the solution is given by the sum of the complimentary and particular functions.
      1. Complimentary function, $y_c$, focuses on the left-side of the equation…
         1. By assuming $y=e^{kx}$, find characteristic equation by substituting into original equation $y$: $y^{\prime }=k{e}^{kx}$ and $y^{\prime \prime }=k^2{e}^{kx}$.
         2. After simplifying, left with a quadratic in $k$: solve.
         3. Final complimentary function $y_c$ is the sum of solutions using the principle of superposition:
            1. Repeated values of $k$: multiply one solution by $x$, i.e. $y_c=y_1+x{y}_1$.
            2. Distinct values of $k$: sum two solutions, i.e. $y_c=y_1+y_2$.
            3. If complex conjugates such that $k=R\pm iQ$ then $y_c=e^{Rx}(c_1\cos (Qx)+c_2\sin (Qx))$, a simplification of distinct values.
      2. Particular function, $y_p$, focuses on the right-side of the equation…
         1. Form of solution is the same as the equation given, i.e.
            1. Polynomial: $\text{RHS}=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$ gives $y_p=A{x}^n+B{x}^{n-1}+\dots +Cx+D$.
            2. Exponential: $\text{RHS}=e^{kx}$ gives $y_p=A{e}^{kx}$.
            3. Sine and Cosine: $\text{RHS}=a_1\sin (kx)+a_2\cos (kx)$ gives $y_p=A\sin (kx)+B\cos (kx)$.
            4. Other Functions: $\text{RHS}=f(x)$ gives $y_p=Af(x)$, considering the domain, or just use variation of parameters.
            5. Combination: Sum of all cases present.
            6. ==IMPORTANT NOTE: If similar terms on both sides of original equation $y$, then multiply by factors of $x$ until distinct to ensure linear independence==.
         2. Differentiate the assumed $y_p$ to give $y^{\prime }$ and $y^{\prime \prime }$ and substitute into original $y$.
         3. Compare coefficients of powers of $x$ between substituted equation and assumed $y_p$ to form simultaneous equations, solved with Gauss-Jordan elimination.
      3. Find, simplifying if possible, $y=y_c+y_p$ as the final solution.
   2. If non-constant coefficients, then the solution can only be found with variation of parameters…
      1. Complimentary function, $y_c$, focuses on the left-side of the equation…
         1. Solutions should be given, unless in the form $a{x}^2\frac{d^{2}x}{d{y}^2}+bx\frac{dy}{dx}+cy=\dots$, a Cauchy-Euler equation.
         2. By assuming $y=x^m$, find characteristic equation by substituting into original equation $y$: $y^{\prime }=m{x}^{m-1}$ and $y^{\prime \prime }=m(m-1)x^{m-2}$.
         3. After simplifying, left with a quadratic in $m$: solve.
         4. Final complimentary function $y_c$ is the sum of solutions using the principle of superposition:
            1. Repeated values of $m$: multiply one solution by $\ln x$, i.e. $y_c=y_1+y_1\ln x$.
            2. Distinct values of $m$: sum two solutions, i.e. $y_c=y_1+y_2$.
            3. If complex conjugates such that $m=R\pm iQ$ then $y_c=x^R(c_1\cos (Q\ln x)+c_2\sin (Q\ln x))$, a simplification of distinct values.
      2. Particular function, $y_p$, focuses on the right-side of the equation…
         1. Ensure equation in standard form $f(x)=\text{RHS}$, if Cauchy-Euler then $f(x)=\frac{\text{RHS}}{x^2}$.
         2. First find the Wronskian $W=W(y_1,y_2)=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }$ using $y_1$, $y_2$ from finding $y_c$.
         3. Then find the variable parameters $u_1=-\int \frac{y_{2}f(x)}{W}$ and $u_2=\int \frac{y_{1}f(x)}{W}$.
         4. Thus particular solution $y_p=u_1{y}_1+u_2{y}_2$. Note: this can be used instead of undetermined coefficients, too, i.e. if a confusing function.
      3. Find, simplifying if possible, $y=y_c+y_p$ as the final solution. [Timestamp: 2024/12/02 18:44:12]

ai: Here’s a reworded version of the differential equations cheat sheet section focusing on the complementary function and the particular function:

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4. Second-Order Solutions:
   1. The general solution to a second-order differential equation is the sum of the complementary function and the particular function.
      1. Complementary Function ($y_c$):
         • The complementary function addresses the homogeneous part of the equation.
         • To find $y_c$, assume a solution of the form $y=e^{kx}$ and substitute into the original equation to derive the characteristic equation:
           • From $y^{\prime }=k{e}^{kx}$ and $y^{\prime \prime }=k^2{e}^{kx}$, simplify to obtain a quadratic in $k$ and solve for $k$.
         • The final complementary function $y_c$ is constructed using the principle of superposition:
           1. For repeated roots of $k$: multiply one solution by $x$, resulting in $y_c=y_1+x{y}_1$.
           2. For distinct roots of $k$: sum the solutions, yielding $y_c=y_1+y_2$.
           3. For complex conjugate roots ($k=R\pm iQ$): the complementary function takes the form $y_c=e^{Rx}(c_1\cos (Qx)+c_2\sin (Qx))$, which simplifies the case of distinct roots.
      2. Particular Function ($y_p$):
         • The particular function addresses the non-homogeneous part of the equation.
         • The form of $y_p$ should match the right-hand side (RHS) of the equation:
           1. For a polynomial RHS: $y_p=A{x}^n+B{x}^{n-1}+\dots +Cx+D$.
           2. For an exponential RHS: $y_p=A{e}^{kx}$.
           3. For sine and cosine terms: $y_p=A\sin (kx)+B\cos (kx)$.
           4. For other functions: $y_p=Af(x)$, considering the domain, or apply variation of parameters.
           5. For a combination of terms: sum all applicable cases.
           6. IMPORTANT NOTE: If terms on both sides of the original equation are similar, multiply by factors of $x$ until they are distinct to ensure linear independence.
         • Differentiate the assumed $y_p$ to obtain $y^{\prime }$ and $y^{\prime \prime }$, then substitute these into the original equation.
         • Compare coefficients of powers of $x$ between the substituted equation and the assumed $y_p$ to form simultaneous equations, which can be solved using methods like Gauss-Jordan elimination.
      3. Finally, combine the results to find the overall solution: $y=y_c+y_p$.
   2. If the coefficients are non-constant, the solution can only be found using variation of parameters:
      1. Complementary Function ($y_c$):
         • Solutions should be derived unless the equation is in the form of a Cauchy-Euler equation: $a{x}^2\frac{d^{2}y}{d{x}^2}+bx\frac{dy}{dx}+cy=\dots$.
         • Assume a solution of the form $y=x^m$, substitute into the original equation to find the characteristic equation: $y^{\prime }=m{x}^{m-1}$ and $y^{\prime \prime }=m(m-1)x^{m-2}$.
         • Simplify to obtain a quadratic in $m$ and solve.
         • The final complementary function $y_c$ is constructed using the principle of superposition:
           1. For repeated roots of $m$: multiply one solution by $\ln x$, resulting in $y_c=y_1+y_1\ln x$.
           2. For distinct roots of $m$: sum the solutions, yielding $y_c=y_1+y_2$.
           3. For complex conjugate roots ($m=R\pm iQ$): the complementary function takes the form $y_c=x^R(c_1\cos (Q\ln x)+c_2\sin (Q\ln x))$, simplifying the case of distinct roots.
      2. Particular Function ($y_p$):
         • Ensure the equation is in standard form $f(x)=\text{RHS}$; if it’s a Cauchy-Euler equation, use $f(x)=\frac{\text{RHS}}{x^2}$.
         • First, compute the Wronskian $W=W(y_1,y_2)=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }$ using the solutions $y_1$ and $y_2$ from the complementary function.
         • Then, find the variable parameters:
           • $u_1=-\int \frac{y_{2}f(x)}{W}dx$ and $u_2=\int \frac{y_{1}f(x)}{W}dx$.
         • The particular solution is then given by $y_p=u_1{y}_1+u_2{y}_2$. Note: This method can also be used instead of undetermined coefficients for complex functions.
      3. Finally, combine the results to find the overall solution: $y=y_c+y_p$.

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This version emphasizes the complementary and particular functions while suggesting appropriate methods for each case. [Timestamp: 2024/12/02 18:45:10]