# Solving Differential Equations by Substitution Methods

### Solving Differential Equations by Substitution Methods:

1. 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.
2. Types of Substitution Methods:

a. Bernoulli Substitution:

• Applicable to first-order nonlinear differential equations in the form $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right){y}^{n}$.
• Substitute $z={y}^{1-n}$ 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.
3. 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.
4. 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 $v\left(x\right)$.

• Differentiate $y=vx$ to find ${y}^{\mathrm{\prime }}$ in terms of $v\left(x\right)$.
• Substitute into the differential equation, solve for $v\left(x\right)$, and eventually obtain $y\left(x\right)$
• 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 $v\left(x\right)$.
• Substitute back the original substitution $y=vx$ to find the solution for $y\left(x\right)$.