Shtokalo studied at Dnipropetrovsk University in south-central Ukraine, beginning his studies there in 1927. It was while he was an undergraduate in the department of physics and mathematics of Dnipropetrovsk University that his first mathematics work was published, namely Handbook of Mathematics for Workers in Heavy Industry (1928). He studied at Dnipropetrovsk University until 1931 but at the same time he had positions teaching mathematics at Vocational Technical Schools and in the Dnepropetrovsk Evening Metallurgical School of the G I Petrovsk Factory. After graduating from Dnipropetrovsk University, he was accepted as a postgraduate student in the Ukrainian Research Institute of Mathematics and Mechanics of Kharkov University in 1931 and he undertook research there for three years on the theory of functions of a complex variable and their applications. He submitted a candidate's thesis (equivalent to a Ph.D.) in 1934; it was on the pressure of a flux of finite width on a plane lamina. After the award of the degree, he was appointed as head of mathematics at the Kharkov Textile Institute and, at the same time, gave mathematics courses at Kharkov University. His research on the theory of functions of a complex variable and its applications included work on conformal mappings, differential equations, and variational statistics. However, during the second half of the 1930s he worked almost exclusively on differential equations.
Although World War II started in September 1939 with the German invasion of Poland, because of the Molotov-Ribbentrop non-aggression pact between Germany and the Soviet Union, this had little effect in Russia and the Ukraine. However, on 22 June 1941 the German armies invaded their former allies pushing rapidly east into Soviet lands. At first their main advance was aimed towards Moscow, but by August they made a strong push in the south deep into the Ukraine capturing Kharkov on 24 October 1941. Shtokalo had been given the task of evacuating the Ukrainian Academy of Sciences from Kharkov to the Urals so by the time Kharkov fell to the Germans, Shtokalo and the Academy had moved far towards the east. An attempt by the Red Army to retake the city of Kharkov from the Germans failed in May 1942. The German occupation of Kharkov lasted until 1943. The first liberation of the city in February 1943 only lasted until March 1943 when again it was captured by the Germans, but the city was finally liberated on 23 August 1943. Shtokalo had moved to Ufa, the capital of the Bashkortostan republic in western Russia, where the Ukrainian Academy of Sciences had been relocated. In Ufa he worked as a senior scientific researcher at the Institute of Mathematics and Physics of the Ukrainian Academy of Sciences and as a lecturer at the Department of Mathematical Analysis headed by Nikolai Nikolaevich Bogolyubov at the Ufa State Aviation Technical University. Shtokalo continued to undertake research on differential equations while in Ufa, but the direction of his research was at this time influenced by the joint work of Bogolyubov with Nikolai Mitrofanovich Krylov in which they developed a theory of non-linear oscillations; they called their topic 'non-linear mechanics'.
In 1943 Shtokalo left Ufa and moved to Moscow where the Institute of Mathematics of the Ukrainian Academy of Sciences had been moved. There he defended his doctoral dissertation (equivalent in level to the D.Sc. or the habilitation) on Asymptotic and Symbolic-Analytic Methods in the Solution of Certain Classes of Linear Differential Equations with Variable Coefficients. In 1944 he also received an appointment as head of the Department of Mathematics at Kiev State University. This was a difficult time following the war when there were not enough textbooks, teaching aids, and stationery for the students. Also many of the students had come straight from the trenches having been drafted as soldiers before completing a good general education. Many had insufficient background to begin their university studies and Shtokalo had to organise the Department to cope with difficulties of this kind. He had a reputation as an excellent lecturer and his teaching skills were of great use in these difficult circumstances. Shtokalo was also greatly assisted by Nikolai Nikolaevich Bogolyubov who worked at Kiev State University from 1946 to 1949.
As examples of Shtokalo's work at this time we note his remarkable publication record. In 1945 several papers appeared including: Méthode asymptotique pour la solution de certaines classes d'équations différentielles linéaires à coefficients variables Ⓣ; Généralisation de la formule fondamentale de la méthode symbolique pour le cas des équations différentielles à coefficients variables Ⓣ; Criteria for stability and instability of the solutions of linear differential equations with quasiperiodic coefficients (Ukrainian, Russian), Linear differential equations of the n th order with quasiperiodic coefficients (Ukrainian, Russian), Systems of linear differential equations with quasiperiodic coefficients (Ukrainian, Russian), and Generalized Gibbs formula for the case of linear differential equations with variable coefficients (Ukrainian, Russian). Then in 1949 he went to Lvov when he became the Chairman of the Lvov Branch of the Ukrainian Academy of Sciences. While there he was involved in local politics being elected to the Lvov City Council.
In 1956 Shtokalo returned to Kiev where he again took up his posts at the Institute of Mathematics of the Ukrainian Academy of Sciences and his chair at Kiev University. At the Institute of Mathematics Shtokalo set up a seminar on the history of mathematics and was head of the Department of History of Mathematics from 1956 to 1963. In 1963, while retaining his position at Kiev University, he moved from the Institute of Mathematics to the Institute of History of the Ukrainian Academy of Sciences in Kiev where he was head of the section on the History of Technology and Science. The held this position until 1986. From 1957 to 1967 he was Chairman of the Terminology Commission of the Presidium of the Ukrainian Academy of Sciences. In 1974 he moved into apartment number 19 in 14/7 Nikolsko Botanica Street in Kiev where he lived until 1987. On 30 November 1989 a bronze plaque was put up on the apartment to mark the place where Shtokalo had lived.
Shtokalo worked mainly in the areas of differential equations, operational calculus and the history of mathematics. Volodymyr Petryshyn, writing in , describes Shtokalo's work:-
After 1945 he became particularly interested in the qualitative and stability theory of solutions of systems of linear ordinary differential equations in the Lyapunov sense and in the 1940s and 1950 published a series of articles and three monographs in these areas.Shtokalo's work had a particular impact on linear ordinary differential equations with almost periodic and quasi-periodic solutions. He extended the applications of the operational method to linear ordinary differential equations with variable coefficients. In 1960 he published On the operational calculus (Russian), giving the following English summary:-
This paper contains a brief outline of investigations in the field of the calculus of operations, indicating the main trends and principal stages of development, as well as a substantiation of the methods of the calculus of operations. The author elucidates the role of native scientists and their contributions in this field of mathematics, which are of great importance both for the development of mathematical theory and for application to urgent problems.In the following year his 128-page monograph Operational methods and their development in the theory of linear differential equations with variable coefficients (Russian) was published. In 1961 the English text Linear differential equations with variable coefficients. Criteria of stability and unstability of their solutions was published being a translation of the Russian book which appeared one year earlier. His 303-page book Operational calculus (Russian) was published in 1972. Hans-Jurgen Glaeske begins a review as follows:-
This book is an introduction to operational calculus, which is based on the theory of the Laplace transformation. It differs from other books in this domain by the consideration of solutions of ordinary differential equations with variable coefficients, based on papers of the author and of K G Valeev.An English translation of this book was published in 1976.
In  Volodymyr Petryshyn describes Shtokalo's work on the history of mathematics:-
He is regarded as one of the founders of the history of Soviet mathematics and particularly of the history in Ukraine and articles about M Ostrogradski and H Voronoy, he edited the three volume collections of Voronoy's (1952-3) and Ostrogradski's works (1959-61), a Russian-Ukrainian mathematical dictionary (1960) and approximately eighteen other Russian-Ukrainian terminology dictionaries.Among the other works he published on the history of mathematics we mention: Achievements in mathematical science in the Academy of Sciences of the URSR during the 30 years of Russian hegemony (Ukrainian) (1947); Development of mathematics in the URSR during the 30 years of Russian hegemony (1948); Works of M V Ostrogradskii on mathematical physics (1952); (with I B Pogrebusskii) The Life and Scientific Pursuits of G F Voronoi (1953); Achievements in mathematics in the T G Shevchenko Kiev State University in the years of Soviet power (1957); (with I B Pogrebusskii) On the works of M V Ostrogradskii on mathematical physics (1959); and (with N I Simonov), Philosophical works of V I Lenin and some questions of the development of mathematics (1960). The authors of  note another of his achievements in the history of mathematics:-
At the initiative of I Z Shtokalo, under his direct editorship, and with his participation as author of several chapters, there has been published the multi-volume fundamental collection of papers, 'History of Indigenous Mathematics' (in five books). He carried out a good deal of the scientific and organizational work for this publication, putting together a potent collection of authors. It should be mentioned that such a publication is the first, not only domestically, but in the world's scientific history literature.Shtokalo has received many honours. He was elected a corresponding member of the USSR Academy of Sciences in 1948 and, three years later, became a full member. In 1965 he was elected to the International Academy of the History of Science, becoming an honorary member of the Academy in 1978. He received the prestigious Koyré award from the International Academy of the History of Sciences in 1970. He was awarded the N M Krylov Prize in 1973. He also received State awards such as Honoured Worker in Science in the USSR (1968) and the Order of Lenin. He was buried at the Baikovoe cemetery in Kiev. We end this biography by quoting the authors of  who gave the following appreciation of Shtokalo on his 70th birthday:-
Teacher in primary, intermediate, and advanced schools, scholar, organiser and director of scientific endeavours, researcher in the area of the theory of differential equations, in the domain of operational calculus, in the area of the history of mathematics, organiser of the publication of works which are classics of indigenous science - such is a far from complete enumeration of the contributions of Iosif Zakharovich Shtokalo to the development of domestic science and culture.
Article by: J J O'Connor and E F Robertson