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Differential Equations · First-Order Equations

Exact Equations — Practice Problems and Worked Solutions

Recover a potential function from differential form.

17 worked problems

Every step is generated and checked by a computer algebra system.

1.
Find the implicit general solution of the exact equation.
$(2 x + y) d x + (x + 2 y) d y = 0$
Answer: $x^{2} + x y + y^{2}$
Step-by-step solution
1. $M_{y} = N_{x} = 1$
2. $F_{x} = M, F_{y} = N$
3. $x^{2} + x y + y^{2} = C$
2.
Find the implicit general solution of the exact equation.
$(3 x^{2} + 2 y) d x + (2 x) d y = 0$
Answer: $x^{3} + 2 x y$
Step-by-step solution
1. $M_{y} = N_{x} = 2$
2. $F_{x} = M, F_{y} = N$
3. $x^{3} + 2 x y = C$
3.
Find the implicit general solution of the exact equation.
$(2 x y) d x + (x^{2} + 2 y) d y = 0$
Answer: $x^{2} y + y^{2}$
Step-by-step solution
1. $M_{y} = N_{x} = 2 x$
2. $F_{x} = M, F_{y} = N$
3. $x^{2} y + y^{2} = C$
4.
Find the implicit general solution of the exact equation.
$(y + e^{x}) d x + (x) d y = 0$
Answer: $x y + e^{x}$
Step-by-step solution
1. $M_{y} = N_{x} = 1$
2. $F_{x} = M, F_{y} = N$
3. $x y + e^{x} = C$
5.
Find the implicit general solution of the exact equation.
$(2 x + 3 y) d x + (3 x + 4 y) d y = 0$
Answer: $x^{2} + 3 x y + 2 y^{2}$
Step-by-step solution
1. $M_{y} = N_{x} = 3$
2. $F_{x} = M, F_{y} = N$
3. $x^{2} + 3 x y + 2 y^{2} = C$
6.
Find the implicit general solution of the exact equation.
$(3 x^{2} y) d x + (x^{3} + 3 y^{2}) d y = 0$
Answer: $x^{3} y + y^{3}$
Step-by-step solution
1. $M_{y} = N_{x} = 3 x^{2}$
2. $F_{x} = M, F_{y} = N$
3. $x^{3} y + y^{3} = C$
7.
Find the implicit general solution of the exact equation.
$(2 x + 2 y) d x + (2 x) d y = 0$
Answer: $x^{2} + 2 x y$
Step-by-step solution
1. $M_{y} = N_{x} = 2$
2. $F_{x} = M, F_{y} = N$
3. $x^{2} + 2 x y = C$
8.
Find the implicit general solution of the exact equation.
$(4 x^{3} + y) d x + (x) d y = 0$
Answer: $x^{4} + x y$
Step-by-step solution
1. $M_{y} = N_{x} = 1$
2. $F_{x} = M, F_{y} = N$
3. $x^{4} + x y = C$
9.
Find the implicit general solution of the exact equation.
$(y) d x + (x + 2 y) d y = 0$
Answer: $x y + y^{2}$
Step-by-step solution
1. $M_{y} = N_{x} = 1$
2. $F_{x} = M, F_{y} = N$
3. $x y + y^{2} = C$
10.
Find the implicit general solution of the exact equation.
$(y + cos (x)) d x + (x) d y = 0$
Answer: $x y + sin (x)$
Step-by-step solution
1. $M_{y} = N_{x} = 1$
2. $F_{x} = M, F_{y} = N$
3. $x y + sin (x) = C$
11.
Find the implicit general solution of the exact equation.
$(3 x^{2} + 3 y) d x + (3 x + 2 y) d y = 0$
Answer: $x^{3} + 3 x y + y^{2}$
Step-by-step solution
1. $M_{y} = N_{x} = 3$
2. $F_{x} = M, F_{y} = N$
3. $x^{3} + 3 x y + y^{2} = C$
12.
Find the implicit general solution of the exact equation.
$(2 x y^{2}) d x + (2 x^{2} y) d y = 0$
Answer: $x^{2} y^{2}$
Step-by-step solution
1. $M_{y} = N_{x} = 4 x y$
2. $F_{x} = M, F_{y} = N$
3. $x^{2} y^{2} = C$
13.
Find the implicit general solution of the exact equation.
$(e^{x}) d x + (2 y) d y = 0$
Answer: $y^{2} + e^{x}$
Step-by-step solution
1. $M_{y} = N_{x} = 0$
2. $F_{x} = M, F_{y} = N$
3. $y^{2} + e^{x} = C$
14.
Find the implicit general solution of the exact equation.
$(5 x^{4} + 2 y) d x + (2 x) d y = 0$
Answer: $x^{5} + 2 x y$
Step-by-step solution
1. $M_{y} = N_{x} = 2$
2. $F_{x} = M, F_{y} = N$
3. $x^{5} + 2 x y = C$
15.
Find the implicit general solution of the exact equation.
$(2 x + y^{3}) d x + (3 x y^{2}) d y = 0$
Answer: $x^{2} + x y^{3}$
Step-by-step solution
1. $M_{y} = N_{x} = 3 y^{2}$
2. $F_{x} = M, F_{y} = N$
3. $x^{2} + x y^{3} = C$
16.
Find the implicit general solution of the exact equation.
$(y - sin (x)) d x + (x) d y = 0$
Answer: $x y + cos (x)$
Step-by-step solution
1. $M_{y} = N_{x} = 1$
2. $F_{x} = M, F_{y} = N$
3. $x y + cos (x) = C$
17.
Find the implicit general solution of the exact equation.
$(2 x + 4 y) d x + (4 x + 6 y) d y = 0$
Answer: $x^{2} + 4 x y + 3 y^{2}$
Step-by-step solution
1. $M_{y} = N_{x} = 4$
2. $F_{x} = M, F_{y} = N$
3. $x^{2} + 4 x y + 3 y^{2} = C$

17 worked problems

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