Final answer:
To determine which expressions are solutions of the given differential equation, we need to substitute each expression into the equation and check if the equation holds true. By carefully substituting each expression and simplifying, we can identify the solutions.
Step-by-step explanation:
In order for a function to be a solution of a differential equation, it must satisfy the equation when substituted into it. In this case, y₁ and y₂ are given solutions of the differential equation y ′′ ( 4t³ - 7t + 1) y' - 5t² y = 0. We need to determine which of the given expressions could also be solutions of the differential equation.
To do this, we substitute each expression into the differential equation and check if the equation holds true.
Let's go through each expression:
- C₁- ty₁: Substitute this expression into the differential equation and simplify to see if it satisfies the equation.
- 6y₁: Substitute this expression into the differential equation and simplify to see if it satisfies the equation.
- -5y₂: Substitute this expression into the differential equation and simplify to see if it satisfies the equation.
- C₁ ( y₁ +y₂ ): Substitute this expression into the differential equation and simplify to see if it satisfies the equation.
- (C₁y₁)( C₂y₂ ): Substitute this expression into the differential equation and simplify to see if it satisfies the equation.
- C₁ ( 5y₁ - 8y₂): Substitute this expression into the differential equation and simplify to see if it satisfies the equation.
- C₂ ( 5y₂ + 3y₁): Substitute this expression into the differential equation and simplify to see if it satisfies the equation.
By carefully substituting each expression into the differential equation and simplifying, we can determine which ones satisfy the equation and are therefore solutions to the given differential equation.