Question

Verify that the given functions are solutions of the corresponding differential equations.

(a) y=Asin2x+Bcos2x;y′′ +4y=0;
(b) y=Ae²ˣ +Be⁻²ˣ ;y′′ −4y=0;
(c) y=xtanx;xy′ =y+x² +y²
(d) y=A² +A/x;y+xy′ =x⁴ (y′)²

Answer 1

To verify if the given functions are solutions of the differential equations, we substitute the functions into the differential equations and check if they satisfy the equations. By substituting each function into its respective differential equation, it can be confirmed that all four functions are solutions.

Step-by-step explanation:

To verify if the given functions are solutions of the corresponding differential equations, we need to substitute the functions into the differential equations and check if they satisfy the equations. Let's go through each function:

(a) y=Asin2x+Bcos2x; y'' +4y=0:

Substituting y=Asin2x+Bcos2x into the differential equation, we have:

y'' +4y = A(-4sin2x) + B(-4cos2x) + A(4sin2x) + B(4cos2x) = 0

Hence, the function y=Asin2x+Bcos2x satisfies the differential equation.

(b) y=Ae²ˣ +Be⁻²ˣ ; y'' −4y=0:

Substituting y=Ae²ˣ +Be⁻²ˣ into the differential equation, we have:

y'' −4y = A(4e²ˣ) + B(-4e⁻²ˣ) - A(4e²ˣ) - B(-4e⁻²ˣ) = 0

Hence, the function y=Ae²ˣ +Be⁻²ˣ satisfies the differential equation.

(c) y=xtanx; xy' =y+x² +y²:

Substituting y=xtanx into the differential equation, we have:

xy' = xsec²x = x(tanx/cos²x) = xtanx = y, and x² + y² = x² + (xtanx)² = x² + x²tan²x = 2x²tan²x = 2xy².

Therefore, xy' = y+x² +y² is satisfied by the function y=xtanx.

(d) y=A² +A/x; y+xy' =x⁴(y')²:

Substituting y=A² +A/x into the differential equation, we have:

y+xy' = (A² + A/x) + x(A/x²) = A² + A/x + A/x = A² + 2A/x = (A² + A/x)(A/x) = (A² + A/x)(y')² = x⁴(y')².

Hence, the function y=A² +A/x satisfies the differential equation.