Solving Differential Equations by Substitution Methods
Solving Differential Equations by Substitution Methods:

Introduction to Substitution Methods:
 Substitution techniques involve substituting variables or functions to transform the original differential equation into a simpler form.
 These methods aim to make the equation more amenable to solving by introducing suitable substitutions that simplify the equation.

Types of Substitution Methods:
a. Bernoulli Substitution:
 Applicable to firstorder nonlinear differential equations in the form $\frac{dy}{dx}+P(x)y=Q(x){y}^{n}$.
 Substitute $z={y}^{1n}$ to transform the equation into a linear form, allowing for easier solution.
b. Power Series Substitution:
 Utilized for solving differential equations by assuming a solution in the form of a power series $y={\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}{x}^{n}$.
 Substituting the power series into the differential equation and solving for coefficients.
c. Trigonometric Substitution:
 Employed for specific equations where trigonometric substitutions like $y=\mathrm{sin}x$ or $y=\mathrm{cos}x$ can simplify the equation.
 Substituting trigonometric functions to transform the equation into a simpler form for solution.

Steps in Substitution Methods:
a. Identify the Type of Differential Equation:
 Recognize the structure of the equation to determine the appropriate substitution method.
b. Select Suitable Substitution:
 Choose a substitution that transforms the equation into a form amenable to solution, based on the equation's characteristics.
c. Perform Substitution and Solve:
 Substitute the chosen variable or function into the differential equation.
 Simplify the equation and solve the resulting transformed equation using integration or other appropriate methods.

Example of Solving a Differential Equation by Substitution:
Consider the differential equation $x{y}^{\mathrm{\prime}}+2y={x}^{3}$.
a. Identify the Equation: $x{y}^{\mathrm{\prime}}+2y={x}^{3}$.
b. Substitution: Substitute $y=vx$ to transform the equation.
c. Solve: Apply the substitution and solve the transformed equation for $v(x)$.
 Differentiate $y=vx$ to find ${y}^{\mathrm{\prime}}$ in terms of $v(x)$.
 Substitute into the differential equation, solve for $v(x)$, and eventually obtain $y(x)$

Example of Solving a Differential Equation by Substitution:
Consider the differential equation $\frac{dy}{dx}=\frac{{x}^{2}+{y}^{2}}{xy}$.
a. Homogenization:
 Multiply both sides by $xy$ to homogenize the equation: $xy\frac{dy}{dx}={x}^{2}+{y}^{2}$.
b. Substitution:
 Substitute $y=vx$ to transform the equation: $x\frac{dv}{dx}+v={v}^{2}+1$.
 This results in a separable equation: $x\frac{dv}{dx}={v}^{2}v+1$.
c. Solve the Separable Equation:
 Separating variables and integrating both sides yields the solution $v(x)$.
 Substitute back the original substitution $y=vx$ to find the solution for $y(x)$.